Anti-Ramsey forbidden poset problems

TL;DR

This paper investigates the maximum number of colors in set family colorings that avoid rainbow copies of posets, establishing asymptotic results for trees and crown posets and connecting to extremal set theory.

Contribution

It introduces and analyzes anti-Ramsey numbers for posets, linking them to extremal set theory and providing asymptotic results for specific classes of posets.

Findings

Asymptotic determination of anti-Ramsey numbers for all tree posets.

Asymptotic results for anti-Ramsey numbers of crown posets.

Connections established between anti-Ramsey and extremal numbers.

Abstract

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$, the $\mathcal{G}$ is a strong copy of $P$. We study the anti-Ramsey numbers $\mathrm{ar}(n,P), \mathrm{ar^*}(n,P)$, the maximum number of colors used in a coloring of $2^{[n]}$ that does not admit a rainbow weak or strong copy of $P$, respectively. We establish connections to the well-studied extremal numbers $\mathrm{La}(n,P)$ and $\mathrm{La^*}(n,P)$ and determine asymptotically $\mathrm{ar^*}(n,T)$ for all tree posets $T$ and $\mathrm{ar^*}(n,O_{2k})$ for all crown posets $O_{2k}$.