On the $d$-cluster generalization of Erd\H{o}s-Ko-Rado

TL;DR

This paper proves Mubayi's conjecture on the maximum size of $d$-cluster-free families in combinatorics, extending the Erd ext{o}s-Ko-Rado Theorem and confirming that the largest such family is a star.

Contribution

It resolves Mubayi's conjecture and establishes a new generalization of the Erd ext{o}s-Ko-Rado Theorem for $d$-clusters.

Findings

Confirmed the maximum size of $d$-cluster-free families as ${n-1 race k-1}$.

Proved that the extremal family is a star.

Completed a significant generalization of the Erd ext{o}s-Ko-Rado Theorem.

Abstract

If $2 \le d \le k$ and $n \ge dk/(d-1)$, a $d$-cluster is defined to be a collection of $d$ elements of ${[n] \choose k}$ with empty intersection and union of size no more than $2k$. Mubayi conjectured that the largest size of a $d$-cluster-free family $\mathcal{F} \subset {[n] \choose k}$ is ${n-1 \choose k-1}$, with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization of the Erd\H{o}s-Ko-Rado Theorem.