Description
Recently, Wang
and Zhang obtained a two-part extension of
the famous Erd\H{o}s-Ko-Rado theorem by determining the maximum size of an
intersecting family $F$ such that every member of $F$
either has $\ell$ elements in $X_1$ or $k$ elements in $X$.
In this talk, I will discuss a related three-part extension of the
Erd\H{o}s-Ko-Rado theorem where we determine maximum size of an intersecting
family $F$ such that each member of $F$ either has $\ell$ elements in $X_1$ or $\ell + 1$ elements in $X - X_i$ for some $i \in {1, 2, 3}$ or $k$ elements in $X$, where $X = X_1 \cup X_2 \cup X_3$ is a partition of $X$ into three parts of equal size. Our methods include the use of Katona's shadow intersection theorem and a recent diversity
theorem of Kupavskii and~Zakharov. I will also discuss some connections with the so-called EKR property of graphs and mention some possible avenues of future research.