2024/10/24 by Rajko Nenadov, Nenadov, Rajko
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2410.18581
openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An n-vertex graph G is locally dense if every induced subgraph of size larger than ζn has density at least d > 0, for some parameters ζ, d > 0. We show that the number of induced subgraphs of G with m vertices and maximum degree significantly smaller than dm is roughly \binomζnm, for m ≪ ζn which is not too small. This generalises a result of Kohayakawa, Lee, Rödl, and Samotij on the number of independent sets in locally dense graphs. As an application, we slightly improve a result of Balogh, Chen, and Luo on the generalised Erdős-Rogers function for graphs with small extremal number.