# On colorings of the Boolean lattice avoiding a rainbow copy of a poset

## TL;DR

This paper investigates colorings of Boolean lattices avoiding rainbow copies of posets, establishing exponential bounds for the size of the smallest color class in large dimensions and introducing a general framework for poset avoidance.

## Contribution

It extends previous results by providing asymptotic bounds for the minimal color class size in rainbow poset avoidance and introduces a general poset coloring framework.

## Key findings

- For fixed k, F(n,k) and f(n,k) grow as 2^{(1/2+o(1))n} for large n.
- Exact values of F(n,3) and f(n,2) are known from recent work.
- Introduces a general approach for avoiding rainbow copies of arbitrary posets P in Boolean lattice colorings.

## Abstract

Let $F(n,k)$ ($f(n,k)$) denote the maximum possible size of the smallest color class in a (partial) $k$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow antichain of size $k$. The value of $F(n,3)$ and $f(n,2)$ has been recently determined exactly. We prove that for any fixed $k$ if $n$ is large enough, then $F(n,k),f(n,k)=2^{(1/2+o(1))n}$ holds. We also introduce the general functions for any poset $P$ and integer $c\ge |P|$: let $F(n,c,P)$ ($f(n,c,P)$) denote the the maximum possible size of the smallest color class in a (partial) $c$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow copy of $P$. We consider the first instances of this general problem.