Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing Kst′ for the subdivision of the bipartite graph Ks,t, we show that ex(nKst′)=O(n3/2−12s). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s,k ≥ 1, we show that ex(nL)=Θ(n1+ssk+1) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(nr) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing Hk for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, Hk−1) = O(n1+1/k−δ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C4 as a subgraph satisfies ex(n, H) = o(n2−1/r).
|Number of pages||30|
|Publication status||Published - 2021 Aug|
Bibliographical noteFunding Information:
Research supported by ERC Consolidator Grant PEPCo 724903. Acknowledgements
Research supported by ERC Starting Grant RanDM 676632.
© 2021, János Bolyai Mathematical Society and Springer-Verlag.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics