Rational Exponents for General Graphs

A rational number r is a realizable exponent for a graph H if there exists a finite family of graphs F such that ex(n,H,F)=Θ(nʳ), where ex(n,H,F) denotes the maximum number of copies of H that an n-vertex F-free graph can have. Results for realizable exponents are currently known only when H is either a star or a clique, with the full resolution of the H=K₂ case being a major breakthrough of Bukh and Conlon. In this paper, we establish the first set of results for realizable exponents which hold for arbitrary graphs H by showing that for any graph H with maximum degree Δ≥ 1, every rational in the interval [v(H)-e(H)/2Δ²,\ v(H)] is realizable for H. We also prove a ``stability'' result for generalized Turán numbers of trees which implies that if T≠ K₂ is a tree with leaves, then T has no realizable exponents in [0,] Z. Our proof of this latter result uses a new variant of the classical Helly theorem for trees, which may be of independent interest.