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arXiv · 2210.08370

A Proof of the $(n,k,t)$-Conjectures

Stacie Baumann·Joseph Briggs
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Abstract

An \emph{$(n,k,t)$-graph} is a graph on $n$ vertices in which every set of $k$ vertices contains a clique on $t$ vertices. Turán's Theorem, rephrased in terms of graph complements, states that the unique minimum $(n,k,2)$-graph is an equitable disjoint union of cliques. We prove that minimum $(n,k,t)$-graphs are always disjoint unions of cliques for any $t$ (despite \allowbreak nonuniqueness of extremal examples), thereby generalizing Turán's Theorem and confirming two conjectures of Hoffman et al.

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Stacie Baumann, Joseph Briggs. 2024-12-21. A Proof of the $(n,k,t)$-Conjectures. https://doi.org/10.37236/12707

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