Rainbow Tur\'an problems for forbidden subposets

TL;DR

This paper investigates the maximum size of set families avoiding rainbow copies of a poset, extending classical forbidden subposet problems to a rainbow setting and establishing asymptotic results for various poset classes.

Contribution

It introduces the concept of rainbow forcing in forbidden subposet problems and determines asymptotics for all tree posets, providing new bounds and connections.

Findings

Established connection between $La^*$ and $La^*_R$ functions via poset rainbow forcing.

Determined asymptotics of $La_R^*(n,T)$ for all tree posets.

Obtained exact or asymptotic results for antichains and complete bipartite posets.

Abstract

A family $\mathcal{G}$ of sets is a copy of a poset $(P,\leqslant)$ if $(\mathcal{G},\subseteq)$ is isomorphic to $(P,\leqslant)$. The forbidden subposet problem asks for determining $La^*(n,P)$, the maximum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$. We study the rainbow version of this problem: what is the maximum size $La_R^*(n,P)$ of a family $\mathcal{F}=\cup_{i=1}^mA^i$ such that all $A^i$ are antichains and there is no copy of $P$ with all sets coming from distinct $A^i$ or equivalently $\mathcal{F}$ admits a proper coloring (sets $F\subset F'$ must receive different colors) with no rainbow copy of $P$. A poset $(Q,\leqslant')$ rainbow forces $(P,\leqslant)$ if any proper coloring $c$ of $Q$ ($q\leqslant' q'$ or $q'\leqslant' q$ implies $c(q)\neq c(q')$) admits a rainbow copy of $P$. We establish connection between the $La^*$ and the $La^*_R$ functions via poset rainbow forcing, determine the asymptotics of $La_R^*(n,T)$ for all tree posets and obtain further exact or asymptotic results for antichains and complete bipartite posets.