A generalization of the Erd\H{o}s-Ko-Rado Theorem

TL;DR

This paper introduces a new upper bound for k-uniform, L-intersecting families of sets, characterizes extremal families, and provides applications including a new proof of the Erdős-Ko-Rado Theorem and improvements on related inequalities.

Contribution

It generalizes the Erdős-Ko-Rado Theorem by establishing bounds for L-intersecting families and characterizing extremal cases, extending classical combinatorial results.

Findings

New upper bound for k-uniform, L-intersecting families

Characterization of extremal families in this setting

Improved bounds for Fisher's inequality and Frankl-Füredi conjecture

Abstract

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson Theorem. As an application, we give a new proof for the Erd\H{o}s-Ko-Rado Theorem, improve Fisher's inequality in the uniform case and give an uniform version of the Frankl-F\"uredi conjecture .