Erd\"{o}s-Ko-Rado Type Theorems and Problems in Set Systems

Jiuqiang Liu 1, 2
1 Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P. R. China
2 Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48917, USA

Abstract     Full Text  PDF

In 1961, Erd\"{o}s, Ko, and Rado proved the following fundamental theorem: Let $n\geq 2k$ and $\mathcal{F}$ be an intersecting family of $k$-subsets of $\{1,2,\ldots,n\}$, then $|\mathcal{F}|\leq \binom{n-1}{k-1}$ with equality only when $\mathcal{F}$ in a star. Since then, many extensions and variations of EKR Theorem have appeared in the literature. We will discuss some of these extensions and variations, including Ray-Chaudhury theorem, Frankl- Wilson Theorem, Frankl-F\"{u}redi Conjecture and other related conjectures.