A Note On The Cross-Sperner Families

Let $(\mathcal{F},\mathcal{G})$ be a pair of families of $[n]$, where $[n]=\{1,2,...,n\}$. If $A\not\subset B$ and $B\not\subset A$ hold for all $A\in\mathcal{F}$ and $B\in\mathcal{G}$, then $(\mathcal{F},\mathcal{G})$ is called a Cross-Sperner pair. P. Frankl and Jian Wang introduced the extremal problem that $m(n)={\rm{max}}\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset2^{[n]}~{\rm{are~cross}}$-${\rm{sperner}}\}$, where $\mathcal{I}(\mathcal{F},\mathcal{G})=\{A\cap B:A\in\mathcal{F},B\in\mathcal{G}\}$. In this note, we prove that $m(n)=2^n-2^{\lfloor\frac{n}{2}\rfloor}-2^{\lceil\frac{n}{2}\rceil}+1$ for all $n>1$. This solves an open problem proposed by P. Frankl and Jian Wang.Comment: We solve an open problem proposed by P. Frankl and Jian Wan

Colouring set families without monochromatic k-chains

A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which $n$-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to $k$-chain-free families. Given a family $\mathcal{F}$ of subsets of $[n]$, we define an $(r,k)$-colouring of $\mathcal{F}$ to be an $r$-colouring of the sets without any monochromatic $k$-chains $F_1 \subset F_2 \subset \dots \subset F_k$. We prove that for $n$ sufficiently large in terms of $k$, the largest $k$-chain-free families also maximise the number of $(2,k)$-colourings. We also show that the middle level, $\binom{[n]}{\lfloor n/2 \rfloor}$, maximises the number of $(3,2)$-colourings, and give asymptotic results on the maximum possible number of $(r,k)$-colourings whenever $r(k-1)$ is divisible by three.Comment: 30 pages, final versio

Improved bounds for cross-Sperner systems

A collection of families (F1,F2,⋯,Fk)∈P([n])k is cross-Sperner if there is no pair i≠j for which some Fi∈Fi is comparable to some Fj∈Fj. Two natural measures of the 'size' of such a family are the sum ∑ki=1|Fi| and the product ∏ki=1|Fi|. We prove new upper and lower bounds on both of these measures for general n and k≥2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011

On some extremal and probabilistic questions for tree posets

Given two posets $P,Q$ we say that $Q$ is $P$-free if $Q$ does not contain a copy of $P$. The size of the largest $P$-free family in $2^{[n]}$, denoted by $La(n,P)$, has been extensively studied since the 1980s. We consider several related problems. Indeed, for posets $P$ whose Hasse diagrams are trees and have radius at most $2$, we prove that there are $2^{(1+o(1))La(n,P)}$ $P$-free families in $2^{[n]}$, thereby confirming a conjecture of Gerbner, Nagy, Patk\'os and Vizer [Electronic Journal of Combinatorics, 2021] in these cases. For such $P$ we also resolve the random version of the $P$-free problem, thus generalising the random version of Sperner's theorem due to Balogh, Mycroft and Treglown [Journal of Combinatorial Theory Series A, 2014], and Collares Neto and Morris [Random Structures and Algorithms, 2016]. Additionally, we make a general conjecture that, roughly speaking, asserts that subfamilies of $2^{[n]}$ of size sufficiently above $La(n,P)$ robustly contain $P$, for any poset $P$ whose Hasse diagram is a tree