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<title>Infinitary combinatorics</title>

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Infinitary combinatorics

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In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in <a href="Combinatorics" title="Combinatorics">combinatorics</a> to <a href="Infinite_set" title="Infinite set">infinite sets</a>.

Some of the things studied include <a href="Continuous_graph" class="mw-redirect" title="Continuous graph">continuous graphs</a> and <a href="Tree_(set_theory)" title="Tree (set theory)">trees</a>, extensions of <a href="Ramsey's_theorem" title="Ramsey's theorem">Ramsey's theorem</a>, and <a href="Martin's_axiom" title="Martin's axiom">Martin's axiom</a>.

Recent developments concern combinatorics of the <a href="Continuum_(set_theory)" title="Continuum (set theory)">continuum</a><a href="#cite_note-1">[1]</a> and combinatorics on successors of singular cardinals.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Ramsey_theory_for_infinite_sets">Ramsey theory for infinite sets</h2></div>

Write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ,\lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa ,\lambda }</annotation>

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</math><img src="./3849c2c1577c33738926be137b6e2365c3a2b737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.728ex; height:2.509ex;" alt="{\displaystyle \kappa ,\lambda }" loading="lazy"> for ordinals, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> for a <a href="Cardinal_number" title="Cardinal number">cardinal number</a> (finite or infinite) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> for a natural number. <a href="#CITEREFErdősRado1956">Erdős & Rado (1956)</a> introduced the notation

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</style><div class="block-indent" style="padding-left: 1.6em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \kappa \rightarrow (\lambda )_{m}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \kappa \rightarrow (\lambda )_{m}^{n}}</annotation>

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</math><img src="./1141683c744d5c179eef184d707952a5bef85cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.793ex; height:2.843ex;" alt="{\displaystyle \displaystyle \kappa \rightarrow (\lambda )_{m}^{n}}" loading="lazy"></div>

as a shorthand way of saying that every <a href="Partition_of_a_set" title="Partition of a set">partition</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\kappa ]^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [\kappa ]^{n}}</annotation>

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</math><img src="./42a0aed1677d3f1b24aa70e6ad61037210d50ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle [\kappa ]^{n}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-element <a href="Subset" title="Subset">subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> pieces has a homogeneous set of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">. A homogeneous set is in this case a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> such that every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-element subset is in the same element of the partition. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is 2 it is often omitted. Such statements are known as partition relations.

Assuming the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, there are no ordinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa \rightarrow (\omega )^{\omega }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \kappa \rightarrow (\omega )^{\omega }}</annotation>

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</math><img src="./02c7484b76484b997f1bbfe58281fc73ecb06e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.463ex; height:2.843ex;" alt="{\displaystyle \kappa \rightarrow (\omega )^{\omega }}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is usually taken to be finite. An extension where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is almost allowed to be infinite is the notation

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \kappa \rightarrow (\lambda )_{m}^{<\omega }}</annotation>

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</math><img src="./391beb5f34a718f8cd2b15dcdb87ebcc9fb6b7e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.651ex; height:2.843ex;" alt="{\displaystyle \displaystyle \kappa \rightarrow (\lambda )_{m}^{<\omega }}" loading="lazy"></div>

which is a shorthand way of saying that every <a href="Partition_of_a_set" title="Partition of a set">partition</a> of the set of finite subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> pieces has a subset of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> such that for any finite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, all subsets of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> are in the same element of the partition. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is 2 it is often omitted.

Another variation is the notation

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \kappa \rightarrow (\lambda ,\mu )^{n}}</annotation>

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</math><img src="./2d7c8b5efc69f69356cf31122146132c29670c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.772ex; height:2.843ex;" alt="{\displaystyle \displaystyle \kappa \rightarrow (\lambda ,\mu )^{n}}" loading="lazy"></div>

which is a shorthand way of saying that every coloring of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\kappa ]^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [\kappa ]^{n}}</annotation>

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</math><img src="./42a0aed1677d3f1b24aa70e6ad61037210d50ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle [\kappa ]^{n}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-element subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> with 2 colors has a subset of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> such that all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\lambda ]^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [\lambda ]^{n}}</annotation>

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</math><img src="./77f4fdb80e903fd56ed442cf20907d757ac29423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.867ex; height:2.843ex;" alt="{\displaystyle [\lambda ]^{n}}" loading="lazy"> have the first color, or a subset of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> such that all elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu ]^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [\mu ]^{n}}</annotation>

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</math><img src="./1d7f9c923824753faef440b78b0816cc4069d9a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.914ex; height:2.843ex;" alt="{\displaystyle [\mu ]^{n}}" loading="lazy"> have the second color. A coloring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\kappa ]^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [\kappa ]^{n}}</annotation>

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</math><img src="./42a0aed1677d3f1b24aa70e6ad61037210d50ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle [\kappa ]^{n}}" loading="lazy">is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:[\kappa ]^{n}\rightarrow p}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:[\kappa ]^{n}\rightarrow p}</annotation>

</semantics>

</math><img src="./598682a44d3563100fab36423a3ad5faa2d0e543.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.85ex; height:2.843ex;" alt="{\displaystyle f:[\kappa ]^{n}\rightarrow p}" loading="lazy">.

Some properties of this include: (in what follows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

</semantics>

</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> is a cardinal)

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{0}\rightarrow (\aleph _{0})_{k}^{n}}</annotation>

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</math><img src="./5ff60f9808071776d4787a9296024df7f1b818ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.591ex; height:3.009ex;" alt="{\displaystyle \displaystyle \aleph _{0}\rightarrow (\aleph _{0})_{k}^{n}}" loading="lazy"> for all finite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> (<a href="Ramsey's_theorem" title="Ramsey's theorem">Ramsey's theorem</a>).</div>

</mrow>

</mrow>

</msubsup>

<msub>

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</msub>

<msub>

</mrow>

</msub>

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</mrow>

</msubsup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \beth _{n}^{+}\rightarrow (\aleph _{1})_{\aleph _{0}}^{n+1}}</annotation>

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</math><img src="./a7cda528a32417f41d06c4b9451b7a2894b88fe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.05ex; width:14.388ex; height:3.676ex;" alt="{\displaystyle \displaystyle \beth _{n}^{+}\rightarrow (\aleph _{1})_{\aleph _{0}}^{n+1}}" loading="lazy"> (the <a href="Erd%C5%91s%E2%80%93Rado_theorem" title="Erdős–Rado theorem">Erdős–Rado theorem</a>.)</div>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle 2^{\kappa }\not \rightarrow (\kappa ^{+})^{2}}</annotation>

</semantics>

</math><img src="./9554db712f5ded14d9994f1e2b144b1b0b98acea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.669ex; height:3.176ex;" alt="{\displaystyle \displaystyle 2^{\kappa }\not \rightarrow (\kappa ^{+})^{2}}" loading="lazy"> (the Sierpiński theorem)</div>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle 2^{\kappa }\not \rightarrow (3)_{\kappa }^{2}}</annotation>

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</math><img src="./4cb7d55e9972e4ddbe793398816dea4cd0bb8c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.106ex; height:3.009ex;" alt="{\displaystyle \displaystyle 2^{\kappa }\not \rightarrow (3)_{\kappa }^{2}}" loading="lazy"></div>

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \kappa \rightarrow (\kappa ,\aleph _{0})^{2}}</annotation>

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</math><img src="./90eb957d510a19e72473537230a9b96eb16ec8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.664ex; height:3.176ex;" alt="{\displaystyle \displaystyle \kappa \rightarrow (\kappa ,\aleph _{0})^{2}}" loading="lazy"> (the <a href="Erd%C5%91s%E2%80%93Dushnik%E2%80%93Miller_theorem" title="Erdős–Dushnik–Miller theorem">Erdős–Dushnik–Miller theorem</a>)</div>

In choiceless universes, partition properties with infinite exponents may hold, and some of them are obtained as consequences of the <a href="Axiom_of_determinacy" title="Axiom of determinacy">axiom of determinacy</a> (AD). For example, <a href="Donald_A._Martin" title="Donald A. Martin">Donald A. Martin</a> proved that AD implies

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\rightarrow (\aleph _{1})_{2}^{\aleph _{1}}}</annotation>

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</math><img src="./29944eb1c69dc1efb7d56d063163aae09f6f16f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.44ex; height:3.509ex;" alt="{\displaystyle \displaystyle \aleph _{1}\rightarrow (\aleph _{1})_{2}^{\aleph _{1}}}" loading="lazy"></div>

<div class="mw-heading mw-heading2"><h2 id="Strong_colorings">Strong colorings</h2></div>

<a href="Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a> showed that the Ramsey theorem does not extend to sets of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>

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</math><img src="./78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}" loading="lazy">by showing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}\nrightarrow (\aleph _{1})_{2}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}\nrightarrow (\aleph _{1})_{2}^{2}}</annotation>

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</math><img src="./eccd5ee402f4596020236ef047e0b2895aa3a98d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.183ex; height:3.343ex;" alt="{\displaystyle 2^{\aleph _{0}}\nrightarrow (\aleph _{1})_{2}^{2}}" loading="lazy">. That is, Sierpiński constructed a coloring of pairs of <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> into two colors such that for every uncountable subset of real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X]^{2}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle [X]^{2}}</annotation>

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</math><img src="./bde891d2facb49a7ba0c0ab91082074137692b35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.328ex; height:3.176ex;" alt="{\displaystyle [X]^{2}}" loading="lazy"> takes both colors. Taking any set of real numbers of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation>

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</math><img src="./78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}" loading="lazy"> and applying the coloring of Sierpiński to it, we get that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}\not \rightarrow (\aleph _{1})_{2}^{2}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{1}\not \rightarrow (\aleph _{1})_{2}^{2}}</annotation>

</semantics>

</math><img src="./c97d776593b2b47761d497892cbc3a8baa60ec34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.427ex; height:3.176ex;" alt="{\displaystyle \aleph _{1}\not \rightarrow (\aleph _{1})_{2}^{2}}" loading="lazy">. Colorings such as this are known as strong colorings<a href="#cite_note-3">[3]</a> and studied in set theory. <a href="#CITEREFErdősHajnalRado1965">Erdős, Hajnal & Rado (1965)</a> introduced a similar notation as above for this.

Write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ,\lambda }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \kappa ,\lambda }</annotation>

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</math><img src="./3849c2c1577c33738926be137b6e2365c3a2b737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.728ex; height:2.509ex;" alt="{\displaystyle \kappa ,\lambda }" loading="lazy"> for ordinals, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> for a cardinal number (finite or infinite) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> for a natural number. Then

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \kappa \nrightarrow [\lambda ]_{m}^{n}}</annotation>

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</math><img src="./ef582d5396054179b1d1e38fac1c5a24c8fdf392.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.277ex; height:2.843ex;" alt="{\displaystyle \displaystyle \kappa \nrightarrow [\lambda ]_{m}^{n}}" loading="lazy"></div>

is a shorthand way of saying that there exists a coloring of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\kappa ]^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [\kappa ]^{n}}</annotation>

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</math><img src="./42a0aed1677d3f1b24aa70e6ad61037210d50ba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.851ex; height:2.843ex;" alt="{\displaystyle [\kappa ]^{n}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-element subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> pieces such that every set of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is a rainbow set. A rainbow set is in this case a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A]^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle [A]^{n}}</annotation>

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</math><img src="./c4e9c84a12e6f13d336de3ba9f8634500033f162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle [A]^{n}}" loading="lazy"> takes all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> colors. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is 2 it is often omitted. Such statements are known as negative square bracket partition relations.

Another variation is the notation

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<annotation encoding="application/x-tex">{\displaystyle \kappa \nrightarrow [\lambda ;\mu ]_{m}^{2}}</annotation>

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</math><img src="./50904feef791286e0814a750092e95cb7a651a19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.713ex; height:3.009ex;" alt="{\displaystyle \kappa \nrightarrow [\lambda ;\mu ]_{m}^{2}}" loading="lazy"></dd></dl>

which is a shorthand way of saying that there exists a coloring of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\kappa ]^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [\kappa ]^{2}}</annotation>

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</math><img src="./144a0e17127cf2f715bb901a7b31dc33c2a1208b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.687ex; height:3.176ex;" alt="{\displaystyle [\kappa ]^{2}}" loading="lazy"> of 2-element subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> colors such that for every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> and every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> of order type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times B}">

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<annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation>

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</math><img src="./65f31ae45b0098f06b5d22c38d317eb097a88fa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}" loading="lazy"> takes all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> colors.

Some properties of this include: (in what follows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

</semantics>

</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> is a cardinal)

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle 2^{\kappa }\nrightarrow [\kappa ^{+}]^{2}}</annotation>

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</math><img src="./08134672050ffeed92707f0c96e407d0452b3516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.153ex; height:3.176ex;" alt="{\displaystyle \displaystyle 2^{\kappa }\nrightarrow [\kappa ^{+}]^{2}}" loading="lazy"> (Sierpiński)</div>

<msub>

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<msub>

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</msub>

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]^{2}}</annotation>

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</math><img src="./e3707146af77544a7f1ff8f8d4523f2f94b2cc33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.911ex; height:3.176ex;" alt="{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]^{2}}" loading="lazy"> (Sierpiński)</div>

<msub>

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<msub>

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</msub>

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</msubsup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{3}^{2}}</annotation>

</semantics>

</math><img src="./431a655b5dfa3455706e7b09acf144f503c18dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.911ex; height:3.176ex;" alt="{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{3}^{2}}" loading="lazy"> (<a href="Richard_Laver" title="Richard Laver">Laver</a>, <a href="Andreas_Blass" title="Andreas Blass">Blass</a>)</div>

<msub>

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</msub>

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</msub>

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</msubsup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{4}^{2}}</annotation>

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</math><img src="./50b35d95134623734b7fd0729e448258d03fd379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.911ex; height:3.176ex;" alt="{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{4}^{2}}" loading="lazy"> (<a href="Fred_Galvin" title="Fred Galvin"> Galvin</a> and <a href="Saharon_Shelah" title="Saharon Shelah">Shelah</a>)</div>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{\aleph _{1}}^{2}}</annotation>

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</math><img src="./f165ee6e3c5278b98eed9bed2fd7a9f6f7cd556d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.925ex; height:3.509ex;" alt="{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1}]_{\aleph _{1}}^{2}}" loading="lazy"> (<a href="Stevo_Todor%C4%8Devi%C4%87" title="Stevo Todorčević">Todorčević</a>)</div>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1};\aleph _{1}]_{\aleph _{1}}^{2}}</annotation>

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</math><img src="./ab7d9064c46dcf299041e084f389d8ff93ba89ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.433ex; height:3.509ex;" alt="{\displaystyle \displaystyle \aleph _{1}\nrightarrow [\aleph _{1};\aleph _{1}]_{\aleph _{1}}^{2}}" loading="lazy"> (<a href="Justin_T._Moore" title="Justin T. Moore">Moore</a>)</div>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle 2^{\aleph _{0}}\nrightarrow [2^{\aleph _{0}}]_{\aleph _{0}}^{2}}</annotation>

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</math><img src="./287852519da52d6ac06263c7344d00d2381ae67f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.437ex; height:3.676ex;" alt="{\displaystyle \displaystyle 2^{\aleph _{0}}\nrightarrow [2^{\aleph _{0}}]_{\aleph _{0}}^{2}}" loading="lazy"> (<a href="Fred_Galvin" title="Fred Galvin"> Galvin</a> and <a href="Saharon_Shelah" title="Saharon Shelah">Shelah</a>)</div>

<div class="mw-heading mw-heading2"><h2 id="Large_cardinals">Large cardinals</h2></div>

Several <a href="Large_cardinal" title="Large cardinal">large cardinal</a> properties can be defined using this notation. In particular:

<ul><li><a href="Weakly_compact_cardinal" title="Weakly compact cardinal">Weakly compact cardinals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

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</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> are those that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa \rightarrow (\kappa )^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa \rightarrow (\kappa )^{2}}</annotation>

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</math><img src="./dcb20ce7d5e7b0e1bbfbf0cf526572d4fd0725ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.156ex; height:3.176ex;" alt="{\displaystyle \kappa \rightarrow (\kappa )^{2}}" loading="lazy"></li>

<li>α-<a href="Erd%C5%91s_cardinal" title="Erdős cardinal">Erdős cardinals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

</semantics>

</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> are the smallest that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa \rightarrow (\alpha )^{<\,\omega }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa \rightarrow (\alpha )^{<\,\omega }}</annotation>

</semantics>

</math><img src="./9a22f76b8aa14c886d5a3b33c995a8614355687d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.17ex; height:3.009ex;" alt="{\displaystyle \kappa \rightarrow (\alpha )^{<\,\omega }}" loading="lazy"></li>

<li><a href="Ramsey_cardinal" title="Ramsey cardinal">Ramsey cardinals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

</semantics>

</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> are those that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa \rightarrow (\kappa )^{<\,\omega }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa \rightarrow (\kappa )^{<\,\omega }}</annotation>

</semantics>

</math><img src="./b0152b8045ad7f14fe31bc5777ecfe65cc78e24f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.022ex; height:3.009ex;" alt="{\displaystyle \kappa \rightarrow (\kappa )^{<\,\omega }}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="Andreas_Blass" title="Andreas Blass">Andreas Blass</a>, Combinatorial Cardinal Characteristics of the Continuum, Chapter 6 in Handbook of Set Theory, edited by <a href="Matthew_Foreman" title="Matthew Foreman">Matthew Foreman</a> and <a href="Akihiro_Kanamori" title="Akihiro Kanamori">Akihiro Kanamori</a>, Springer, 2010

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Todd Eisworth, Successors of Singular Cardinals Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010

</li>

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</style><cite id="CITEREFRinot" class="citation cs2">Rinot, Assaf, <a rel="nofollow" class="external text" href="https://blog.assafrinot.com/?p=4392">Tutorial on strong colorings and their applications, 6th European Set Theory Conference</a>, retrieved 2023-12-10</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFDushnikMiller1941" class="citation cs2">Dushnik, Ben; Miller, E. W. (1941), "Partially ordered sets", <a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a>, 63 (3): 600–610, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371374">10.2307/2371374</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F100377">10338.dmlcz/100377</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371374">2371374</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0004862">0004862</a></cite></li></ul>

<ul><li><cite id="CITEREFErdősHajnalRado1965" class="citation cs2"><a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős, Paul</a>; <a href="Andr%C3%A1s_Hajnal" title="András Hajnal">Hajnal, András</a>; <a href="Richard_Rado" title="Richard Rado">Rado, Richard</a> (1965), <a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01886396">"Partition relations for cardinal numbers"</a>, Acta Math. Acad. Sci. Hung., 16 (1–2): 93–196, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01886396">10.1007/BF01886396</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0202613">0202613</a></cite></li></ul>

<ul><li><cite id="CITEREFErdősHajnal1971" class="citation cs2 cs1-prop-long-vol"><a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős, Paul</a>; <a href="Andr%C3%A1s_Hajnal" title="András Hajnal">Hajnal, András</a> (1971), "Unsolved problems in set theory", Axiomatic Set Theory ( Univ. California, Los Angeles, Calif., 1967), Proc. Sympos. Pure Math, vol. XIII Part I, Providence, R.I.: Amer. Math. Soc., pp. 17–48, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0280381">0280381</a></cite></li>

<li><cite id="CITEREFErdősHajnalMátéRado1984" class="citation cs2"><a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős, Paul</a>; <a href="Andr%C3%A1s_Hajnal" title="András Hajnal">Hajnal, András</a>; Máté, Attila; <a href="Richard_Rado" title="Richard Rado">Rado, Richard</a> (1984), Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, Amsterdam: North-Holland Publishing Co., <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-86157-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0795592">0795592</a></cite></li>

<li><cite id="CITEREFErdősRado1956" class="citation cs2"><a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős, P.</a>; <a href="Richard_Rado" title="Richard Rado">Rado, R.</a> (1956), <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-5/A-partition-calculus-in-set-theory/bams/1183520996.pdf">"A partition calculus in set theory"</a> (PDF), Bull. Amer. Math. Soc., 62 (5): 427–489, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1956-10036-0">10.1090/S0002-9904-1956-10036-0</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0081864">0081864</a></cite></li>

<li><cite id="CITEREFKanamori,_Akihiro2000" class="citation cs2"><a href="Akihiro_Kanamori" title="Akihiro Kanamori">Kanamori, Akihiro</a> (2000), <a href="The_Higher_Infinite" title="The Higher Infinite">The Higher Infinite</a> (second ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-00384-3</bdi></cite></li>

<li><cite id="CITEREFKunen1980" class="citation cs2"><a href="Kenneth_Kunen" title="Kenneth Kunen">Kunen, Kenneth</a> (1980), <a href="Set_Theory%3A_An_Introduction_to_Independence_Proofs" title="Set Theory: An Introduction to Independence Proofs">Set Theory: An Introduction to Independence Proofs</a>, Amsterdam: North-Holland, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-85401-8</bdi></cite></li></ul></div><div><div class="zim-footer">

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