An Intersection-Weighted Erd\H{o}s-Ko-Rado Theorem

TL;DR

This paper introduces a weighted sum version of the Erdős-Ko-Rado theorem, proving an upper bound for large sets and extending related theorems on intersecting families.

Contribution

It generalizes the Erdős-Ko-Rado theorem to a weighted sum context for large sets and extends Hilton's theorem on cross-intersecting families.

Findings

Weighted sum is at most one for large ground sets.

Equality holds precisely for star families.

Generalizes Erdős-Ko-Rado theorem for all intersection thresholds.

Abstract

We consider an Erd\H{o}s-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with equality exactly for stars. This simultaneously generalizes the usual Erd\H{o}s-Ko-Rado theorem for every intersection threshold $t$ and $n$ sufficiently large. As a consequence we also obtain an extension of Hilton's theorem on cross-intersecting families.