Work

Maximizing $K_r + I_r$ in graphs with fixed edge density

Year: 2026

Type: preprint

Abstract: For every integer $r\ge4$, and $ρ\in [0,1]$, we asymptotically determine the maximum proportion of $r$-element sets of vertices that induce either a clique or an independent set in a large graph with density $ρ$. This generalizes a result of Olpp for $r=3$. After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.08194

Authors: Balogh, József, Grzesik, Andrzej, Lidický, Bernard, Molla, Theodore, Mubayi, Dhruv, Volec, Jan

Institutions: University of Illinois Urbana-Champaign, Jagiellonian University, Iowa State University, University of South Florida, University of Illinois Chicago, Czech Technical University in Prague

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, American Institute of Mathematics, Science Foundation Ireland, Ministerstvo Školství, Mládeže a Tělovýchovy, Grantová Agentura České Republiky, Institute for Basic Science, University of Illinois at Urbana-Champaign

Awards: FRG: Collaborative Research: Extremal Combinatorics and Flag Algebras, RTG: Research in Combinatorics, G2765210714, Global Structures in Graphs and Directed Graphs, G5389903929, FRG: Collaborative Research: Extremal Combinatorics and Flag Algebras, G7055633160, Hypergraphs and Ramsey Theory

OpenAlex ID: W7212015675

Cites: 12

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Related to: 10