Equivalences of promise compactness principles

TL;DR

This paper explores the logical equivalence between certain promise compactness principles related to homomorphisms of finite relational structures and the ultrafilter principle within ZF set theory.

Contribution

It establishes that specific promise principles are equivalent to the ultrafilter principle when the structures lack Olšák polymorphisms, linking combinatorial properties to foundational set theory.

Findings

$K_{(rak{A},rak{B})}$ is equivalent to the ultrafilter principle over ZF.

Includes specific cases like $K_{(K_3,K_5)}$ and $K_{(H_2,H_c)}$ for all $c\, extgreater= 2$.

In ZF models, finite colorability implies infinite colorability for certain graphs.

Abstract

For a pair of finite relational structures $(\mathfrak{A},\mathfrak{B})$ such that $\mathfrak{A}$ homomorphically maps to $\mathfrak{B}$ we denote by $K_{(\mathfrak{A},\mathfrak{B})}$ the following statement: for all structures $\mathfrak{I}$ with the same signature as $\mathfrak{A}$ if all finite substructures of $\mathfrak{I}$ homomorphically maps to $\mathfrak{A}$ then $\mathfrak{I}$ homomorphically maps to $\mathfrak{B}$. In this article, we show that if $(\mathfrak{A},\mathfrak{B})$ has no Ol\v{s}\'{a}k polymorphism, then $K_{(\mathfrak{A},\mathfrak{B})}$ is equivalent to the ultrafilter principle over $\operatorname{ZF}$. This includes the statements $K_{(K_3,K_5)}$ and $K_{(H_2,H_c)}$ for all $c\geq 2$ where $K_n$ denotes the clique of size $n$ and $H_k$ denotes the ternary not-all-equal structure on a $k$-element set. This means, for example, that in any $\operatorname{ZF}$ model, if every finitely 3-colourable graph can be coloured by 5 colours then all these graphs can in fact be coloured by 3 colours.