A simple proof of Talbot's theorem for intersecting separated sets

TL;DR

This paper presents a concise proof of Talbot's theorem, establishing an optimal upper bound on the size of intersecting families of k-separated r-element subsets within a finite set, under certain conditions.

Contribution

The paper provides a new, simplified proof of Talbot's theorem, enhancing understanding of intersecting separated sets and their maximum family sizes.

Findings

Proves the maximum size of intersecting k-separated families is binom{n - kr - 1}{r - 1}

The bound is shown to be tight and optimal

Simplifies the proof of a known combinatorial result

Abstract

A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets is intersecting if every two sets in $\mathcal{A}$ intersect. We give a short and simple proof of a remarkable result of Talbot (2003), stating that if $n \geq (k + 1)r$ and $\mathcal{A}$ is an intersecting family of $k$-separated $r$-element subsets of $[n]$, then $|\mathcal{A}| \leq \binom{n - kr - 1}{r - 1}$. This bound is best possible.