Ramsey functions, property B, and the Axiom of Choice

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$ (the first infinite ordinal) and let $c:[\omega]^\omega\to \{0,1\}$ be any map. We say that $a \in [\omega]^\omega$ is monochromatic with respect to $c$ if $c$ restricted to ${\mathcal P}(a)\cap [\omega]^\omega$ is constant (or, equivalently, if $c(b) = c(a)$ for all $b\in [\omega]^\omega$ with $b\subseteq a$ ). We say $c: [\omega]^\omega\to \{0,1\}$ is Ramsey if there is $a\in [\omega]^\omega$ such that $a$ is monochromatic with respect to $c$ .

Certainly, plenty of examples of Ramsey functions $c:[\omega]^\omega\to\{0,1\}$ come to mind, which leads to the question:

Are there non-Ramsey functions $c: [\omega]^\omega\to\{0,1\}$ ?

It is indeed a bit tricky to come up with one – the construction I present next needs the Axiom of Choice ( https://en.wikipedia.org/wiki/Axiom_of_choice ). We define a binary relation on $[\omega]^\omega$ by

$a\simeq_{\text{fin}}$ if and only if the symmetric difference $a \;\Delta\; b = (a\setminus b) \cup (b\setminus a)$ is finite.

From “the infinitist’s point of view” two sets are the same if they only differ in finitely many elements. It is easy to check that this an equivalence relation.

Let $[\omega]^\omega/\simeq_{\text{fin}}$ be the set of equivalence classes of $[\omega]^\omega$ with respect to the equivalence relation $\simeq_{\text{fin}}$ . We use the Axiom of Choice to find a set of representatives $R\subseteq [\omega]^\omega$ such that

$|R \cap b| = 1$ for all blocks $b\in [\omega]^\omega/\simeq_{\text{fin}}$ .

We define the representative function $r:[\omega]^\omega\to [\omega]^\omega$ by assigning to each $a\in [\omega]^\omega$ the unique element $r(a)\in R$ such that $a\simeq_{\text{fin}} r(a)$ .

Finally, we define $c:[\omega]^\omega\to\{0,1\}$ by setting $c(a) = 0$ for $a\in [\omega]^\omega$ if $|a\;\Delta\; r(a)|$ is even, and $c(a) = 1$ otherwise. It is easy to see that whenever $a\in [\omega]^\omega$ , then $a$ is not monochromatic: As $a \;\Delta\; r(a)$ is finite and $a$ and $r(a)$ are both infinite, pick $n_0\in a\cap r(a)$ . So the parity of

$|a\;\Delta\; r(a)|\;$ and $\;|(a\setminus\{n_0\}) \;\Delta\; r(a)|$

is not the same, therefore $c$ restricted to ${\mathcal P}(a)\cap [\omega]^\omega$ is not constant.

Reformulation with property ${\mathbf B}$

A hypergraph is a pair $H=(V,E)$ where $V$ is a non-empty set and $E\subseteq {\mathcal P}(V)$ . We say that $H$ has property $\mathbf{B}$ if there is $S\subseteq V$ such that $S\cap e\neq\emptyset\neq (e\setminus S)$ for all $e\in E$ . (In other words, $S$ intersects every $e\in E$ , but no $e\in E$ is fully contained in $S$ .)

The question whether there is a non-Ramsey function $c: [\omega]^\omega \to \{0,1\}$ is equivalent to asking whether the hypergraph $H_{\text{Ramsey}} = ([\omega]^\omega, E)$ where

$E = \{{\mathcal P}(a)\cap [\omega]^\omega: a\in [\omega]^\omega\}$

has property $\mathbf{B}$ .

Further questions

Does the statement “ $H_{\text{Ramsey}}$ has property $\mathbf{B}$ ” imply the Axiom of Choice? Or does it imply a weaker statement than AC? Or is it provable in ZF alone?
Consider the hypergraph $H = ([\omega]^\omega/\simeq_{\text{fin}}, E)$ where $E$ is built in an analogous manner to the edge set in hypergraph $H_{\text{Ramsey}}$ . Does $H$ have property ${\mathbf B}$ , no matter whether we assume the Axiom of Choice?
What is in ${\sf (ZF)}$ an example of a poset $(P,\leq)$ with no minimal elements such that the hypergraph $H_P = (P, E)$ does not have property ${\mathbf B}$ , where $E$ is the collection of principal down-sets of $P$ ? (In other words, $(P, \leq)$ has the property that whenever its elements are colored red or blue, then there is a principal down-set consisting of more than 1 element such that all the elements have the same color.)