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

TL;DR

This work resolves the Strong $(n,k,t)$-Conjecture by proving that every minimum $(n,k,t)$-graph is a disjoint union of cliques for all positive integers $n\ge k\ge t$. The authors develop an inductive approach on $t$ and introduce $(n,k,t,r)$-graphs with fixed independence number $\alpha(G)=r$ to track structure, using a Turán-based comparison graph and a detailed edge-count argument to force clique unions. The result generalizes Turán’s theorem (the case $t=2$) and confirms conjectures of Hoffman et al., providing an exact, non-asymptotic description and a framework adaptable to related extremal questions such as clique counts and saturation. Overall, the paper advances the understanding of extremal graph structures by delivering a robust, inductive methodology that yields a complete structural characterization of minimum $(n,k,t)$-graphs and lays groundwork for further combinatorial investigations.

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.

Figures (1)

• Figure 1:

Theorems & Definitions (20)

• Definition 1.1
• Conjecture 1.2: The Weak $(n,k,t)$-Conjecture
• Conjecture 1.3: The Strong $(n,k,t)$-Conjecture
• Theorem 1.4
• Theorem 2.1: Turán's Theorem
• Theorem 2.2: Hoffman
• Example 2.3
• proof
• Lemma 3.1
• proof