On the sizes of $t$-intersecting $k$-chain-free families

TL;DR

This paper establishes the maximum size of $t$-intersecting $k$-Sperner families within the Boolean lattice for large even $n+t$, showing the bound is tight and identifying open cases for odd $n+t$.

Contribution

It proves a tight upper bound on the size of $t$-intersecting $k$-Sperner families for large even $n+t$, extending combinatorial extremal set theory.

Findings

Maximum size matches the sum of $k$ layers centered at $(n+t)/2$

Bound is tight and achieved by specific layered families

Open problem remains for odd $n+t$ cases

Abstract

A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length $k+1$. Our main result is the following: Suppose that $k$ and $t$ are fixed positive integers, where $n+t$ is even with $t\le n$ and $n$ is large enough. If $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then $|\mathcal{F}|$ has size at most the size of the sum of $k$ layers, of sizes $(n+t)/2,\ldots, (n+t)/2+k-1$. This bound is best possible. The case when $n+t$ is odd remains open.