A note on traces of set families

Abstract

A family of sets $\mathcal{F}\subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\mathcal{F}{|}_L=\{F\cap L:F\in \mathcal{F}\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l^{\prime },k$ with $l^{\prime }<k$ if $\mathcal{F}$ is $(n-l^{\prime })$-trace $k$-Sperner, then $|\mathcal{F}|\le (k-l^{\prime }+o(1))\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ and this bound is asymptotically tight.

Cited by in corpus (4)

• Tight paths in convex geometric hypergraphs
• Kalai's conjecture in $r$-partite $r$-graphs
• On tight cycles in hypergraphs
• Forbidden subposet problems for traces of set families