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Set systems containing no singleton intersection and the Delsarte number

William Linz

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 Lovász number and furnishes an infinite family of graphs where the Schrijver variant of the Lovász number is strictly smaller than the Lovász 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.

Set systems containing no singleton intersection and the Delsarte number

Abstract

We prove that the maximum size of a family of -element subsets of the set which contains no singleton intersection is when . This improves upon a recent result of Cherkashin. Our proof uses Schrijver's variant of the Lovász number and furnishes an infinite family of graphs where the Schrijver variant of the Lovász number is strictly smaller than the Lovász number. As a consequence of our result and a recent result of Keller and Lifshitz, it follows that for sufficiently large, the maximum size of a -uniform family on containing no singleton intersection is for all , which is the best possible threshold.

Paper Structure

This paper contains 3 sections, 12 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The Delsarte number and the proof of Theorem \ref{['thm:nosingletonint']}
  3. Concluding remarks

Key Result

Theorem 1

Let $k\ge 4$. Let $\mathcal{F} \subset \binom{[n]}{k}$ be a $(n, k, \{0, 2, \ldots, k-1\})$-system. Then there exists an $n_0(k)$ such that for $n\ge n_0(k)$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (17)

  • Definition 1: $(n, k, L)$-system
  • Definition 2: Generalized Johnson graph
  • Theorem 1: Frankl
  • Theorem 2: Cherkashin
  • Theorem 3
  • Theorem 4: Keller--Lifshitz
  • Corollary 1
  • Theorem 5: Sandwich theoremLovasz1979
  • Theorem 6: Schrijver
  • Lemma 1
  • ...and 7 more