Rainbow Turán problems for forbidden subposets

TL;DR

This work studies the rainbow version of the forbidden subposet problem via $La_R^*(n,P)$ and connects it to the classical $La^*(n,P)$ through rainbow forcing, introducing the minimal forcing posets $M(P)$ with $La_R^*(n,P)=La^*(n,M(P))$. It develops a blow-up framework showing that certain $P^\pi$ rainbow-force $P$, and establishes universal bounds $m(P)\le \perp(P)\le \binom{|P|+1}{2}$, with $m(A_k)=\binom{k+1}{2}$. Using these tools, the authors obtain the tree-poset asymptotics $La_R^*(n,T)=(|T|-1+o(1))\binom{n}{\lfloor n/2\rfloor}$ and determine asymptotics or tight bounds for multipartite and antichain families such as $A_k$, $K_{s,t}$, and the diamond poset $\Diamond$, while introducing a universal tree poset $\mathcal{T}^k$ that rainbow-forces all tree posets on $k$ elements. The results bridge rainbow Turán-type methods with classical forbidden subposet theory and point to graph analogues and several open questions about the structure and finiteness of $M(P)$.

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.

Figures (1)

• Figure 1: Hasse diagram of $D^j_k$.

Theorems & Definitions (39)

• Proposition 1.1
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5: Boehnlein, Jiang BJ
• Theorem 1.6
• Proposition 1.7
• Theorem 1.8
• Remark 1.9
• Lemma 2.1
• proof