Erd\H{o}s-Gy\'{a}rf\'{a}s problem for partially ordered sets

TL;DR

This paper investigates a Boolean lattice coloring problem related to the Erdős-Gyárfás problem, establishing existence results, probabilistic bounds, and lower bounds for the minimum number of colors needed.

Contribution

It proves the existence of strong Boolean Ramsey numbers for posets and provides bounds for the Boolean lattice coloring function $f_t^{lat}(n,p,q)$.

Findings

Existence of strong Boolean Ramsey numbers for all finite posets.

Probabilistic upper bounds on the coloring function.

Lower bounds derived from extremal and combinatorial methods.

Abstract

Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its $t$-chains. Let $f_t^{\sharp}(n,p,q)$ denote the minimum number of colors in such a coloring. We study this Boolean-lattice analogue of the Erd\H{o}s-Gy\'{a}rf\'{a}s function.We first show that every finite poset strongly embeds into a Boolean lattice. Combined with a structural Ramsey theorem for finite posets with linear extensions, this implies the existence of the strong Boolean Ramsey number $\mathrm{R}^{\sharp}_{k,t}(\mathcal{B}\mid Q)$ for every integer $k\ge1$, every $t\ge1$, and every nonempty finite poset $Q$. In particular, this gives an affirmative answer to a problem of Cox and Stolee and yields the existence of $f_t^{\sharp}(n,p,2)$. Next, using the symmetric Lov\'asz local lemma, we obtain a probabilistic upper bound on $f_t^{\sharp}(n,p,q)$. Finally, we prove lower bounds by combining Tur\'an-type extremal estimates for $t$-chains, a double-counting argument, and a generalized Lubell-type framework for $t$-chains.