Short proofs of three results about intersecting systems

TL;DR

This paper presents concise proofs of three theorems related to intersecting systems, including bounds on intersecting families, a version of the Erd ext{"o}s-Ko-Rado theorem, and graph intersection constructions, advancing understanding in combinatorics.

Contribution

It provides simplified proofs of key theorems in intersection theory, extends results to $d$-wise, $t$-intersecting families, and constructs larger graph-intersecting families, addressing open conjectures.

Findings

Determined maximum size of certain intersecting families for specific parameters.

Extended Erd ext{"o}s-Ko-Rado theorem to $d$-wise, $t$-intersecting families.

Constructed larger $K_{s,t}$-intersecting graph families for large $t$.

Abstract

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$, which improves upon a recent result of O'Neill and Verstra\"{e}te. Our proof also extends to $d$-wise, $t$-intersecting families, and from this result we obtain a version of the Erd\H{o}s-Ko-Rado theorem for $d$-wise, $t$-intersecting families. The second result partially proves a conjecture of Frankl and Tokushige about $k$-uniform families with restricted pairwise intersection sizes. The third result concerns graph intersections. Answering a question of Ellis, we construct $K_{s, t}$-intersecting families of graphs which have size larger than the Erd\H{o}s-Ko-Rado-type construction whenever $t$ is sufficiently large in terms of $s$.