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

TL;DR

This paper proves that minimum $(n,k,t)$-graphs are always disjoint unions of cliques, generalizing Turán's Theorem and confirming two conjectures, thus advancing understanding of graph structures with clique constraints.

Contribution

It establishes that minimum $(n,k,t)$-graphs are disjoint unions of cliques for any $t$, extending Turán's Theorem and resolving two open conjectures.

Findings

Minimum $(n,k,t)$-graphs are disjoint unions of cliques.

Generalization of Turán's Theorem to arbitrary $t$.

Confirmation of two conjectures by Hoffman et al.

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. Tur\'an'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 Tur\'an's Theorem and confirming two conjectures of Hoffman et al.