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A family of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph G, let \({\mu }(G)\) denote the size of the smallest maximal independent set of G. In 2005, Holroyd and Talbot conjectured the following generalization of the Erdős-Ko-Rado Theorem: for \(1\le r\le {\mu }(G)/2\) , there is a maximum size intersecting family of independent r-sets that is a star. In this paper we present the history of this conjecture and survey the results that have supported it over the last 20 years.

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A Survey of the Holroyd-Talbot Conjecture

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  • 发表日期：

    2026年

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  • Glenn Hurlbert

摘要

A family of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph G, let \({\mu }(G)\) denote the size of the smallest maximal independent set of G. In 2005, Holroyd and Talbot conjectured the following generalization of the Erdős-Ko-Rado Theorem: for \(1\le r\le {\mu }(G)/2\) , there is a maximum size intersecting family of independent r-sets that is a star. In this paper we present the history of this conjecture and survey the results that have supported it over the last 20 years.