More on the extremal number of subdivisions

Abstract

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 K′_(s,t) for the subdivision of the bipartite graph K_(s,t), we show that ex(n,K′_(s,t)) = O(n^(3/2)−1/2s)). 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(n,L) = Θ(n^(1+s/sk+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(n^r) 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 H^k 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, H^(k−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(n^(2−1/r)).

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