2019/03/25 by Conlon, David, Janzer, Oliver, Lee, Joonkyung · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.10631
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 Ks,t, we show that ex(n, K's,t) = O(n3/2 - (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(n1 + (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(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).