The Kohayakawa–Nagle–Rödl-Schacht conjecture roughly states that every sufficiently large locally d-dense graph G on n vertices must contain at least (1 − o(1))d|E(H)|n|V (H)| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.
|Number of pages||23|
|Journal||Random Structures and Algorithms|
|Publication status||Published - 2021 Mar|
Bibliographical noteFunding Information:
I would like to thank David Conlon for bringing Conjecture 1.1 to my attention and for many helpful discussions. I am also grateful to Oliver Riordan, Mathias Schacht, and Sasha Sidorenko, who carefully read various versions of this paper and gave useful comments. I would like to thank anonymous referees for careful reports that improved the presentation of the draft, too.
This research was supported by the European Research Council, ERC Consolidator Grant, PEPCo 724903. Funding information
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All Science Journal Classification (ASJC) codes
- Computer Graphics and Computer-Aided Design
- Applied Mathematics