2026/07/21 by Tao Jiang, Sean Longbrake, Liana Yepremyan
#math.CO
Given a graph H, the extremal number ex(n,H) is the maximum number of edges in an n-vertex graph not containing H as a subgraph. The well-known rational exponents conjecture of Erdős and Simonovits states that for any rational γ∈ (1,2) there exists a single bipartite graph H satisfying ex(n,H)=Θ(nγ). Among other results, the conjecture has been verified for all γ=1+a/b, where b>a2, by Jiang and Qiu and for all γ=2-a/b, where b>max\a, (a-1)2\, by Conlon and Janzer. In this paper, we establish the rational exponents conjecture for many γ near the center of the interval, namely, for all γ=1+(rt-1)/(2rt+2r), where r,t are natural numbers satisfying t≥ 2, r≥ 2t+3.