Set systems containing no singleton intersection and the Delsarte number

TL;DR

This paper determines the maximum size of certain set families with no singleton intersections, improving previous bounds and exploring graph invariants related to the Lovász number.

Contribution

It proves a new maximum size bound for set families with no singleton intersection and analyzes Schrijver's Lovász variant, extending known results.

Findings

Maximum size of set families is inom{n-2}{k-2}or specified n ranges.

Schrijver's Love1sz number can be strictly smaller than the Love1sz number.

For large k, the maximum size bound holds for all n t least 3k-3fter recent improvements.

Abstract

We prove that the maximum size of a family of $k$-element subsets of the set $[n] = \{1, 2, \ldots, n\}$ which contains no singleton intersection is $\binom{n-2}{k-2}$ when $3k-3 \le n \le k^2-k+1$. This improves upon a recent result of Cherkashin. Our proof uses Schrijver's variant of the Lov\'asz number and furnishes an infinite family of graphs where the Schrijver variant of the Lov\'asz number is strictly smaller than the Lov\'asz number. As a consequence of our result and a recent result of Keller and Lifshitz, it follows that for $k$ sufficiently large, the maximum size of a $k$-uniform family on $[n]$ containing no singleton intersection is $\binom{n-2}{k-2}$ for all $n\ge 3k-3$, which is the best possible threshold.