Kohayakawa-Nagle-R{\"o}dl-Schacht conjecture for subdivisions

TL;DR

This paper advances understanding of the KNRS conjecture by proving that certain subdivisions of graphs, especially balanced subdivisions of cliques, satisfy the conjecture, thus extending previous results in extremal graph theory.

Contribution

It establishes that subdivisions of graphs satisfying the KNRS conjecture also satisfy related conjectures, including Sidorenko's conjecture, especially for regular and balanced cases.

Findings

Subdivisions of graphs satisfying KNRS also satisfy Sidorenko's conjecture.

Balanced subdivisions of cliques satisfy the KNRS conjecture.

Regular graphs' subdivisions preserve the KNRS property.

Abstract

In this paper, we study the well-known Kohayakawa-Nagle-R{\"o}dl-Schacht (KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS conjecture asserts that for any graph $H$, locally dense graphs contain asymptotically at least the number of copies of $H$ found in a random graph with the same edge density. We prove the following results about $k$-subdivisions of graphs (obtained by replacing edges with paths of length $k+1$): (1). If $H$ satisfies the KNRS conjecture, then its $(2k-1)$-subdivision satisfies Sidorenko's conjecture, extending a prior result of Conlon, Kim, Lee and Lee; (2). If $H$ satisfies the KNRS conjecture, then its $2k$-subdivision satisfies a constant-fraction version of the KNRS conjecture; (3). If $H$ is regular and satisfies the KNRS conjecture, then its $2k$-subdivision also satisfies the KNRS conjecture. These findings imply that all balanced subdivisions of cliques satisfy the KNRS conjecture, improving upon a recent result of Brada\v{c}, Sudakov and Wigerson. Our work provides new insights into this pivotal conjecture in extremal graph theory.