Kohayakawa-Nagle-R\"odl-Schacht conjecture for subdivisions

In this paper, we study the well-known Kohayakawa-Nagle-R\"odl-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 c, Sudakov and Wigerson. Our work provides new insights into this pivotal conjecture in extremal graph theory.