Description
Abstract:
A set system $F$ is $t-{\it intersecting}$, if the size of the intersection of every pair of its elements has size at least $t$.
A set system $F$ is $k-{\it Sperner}$, 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 $F\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then $|F|$ 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.
Joint work with Józsi Balogh and Will Linz.
The lecture can be followed by zoom:
https://us06web.zoom.us/j/81239725712?pwd=eDlXNXR1cUhVbWRNbUdEc1dEcWo2dz09
If it is needed, the passcode is 670457.