Negligible Obstructions and Turán Exponents

Abstract

We show that for every rational number $r∈(1,2)$ of the form $2−a/b,$ where $a, b∈\mathbb{N}^+$ satisfy $$\lfloor b/a\rfloor ^3 ≤a≤b/(\lfloor b/a\rfloor +1)+1,$$ there exists a graph $F_r$ such that the Turán number ${\rm ex}(n,F_r)=Θ(n^r).$ Our result in particular generates infinitely many new Turán exponents. As a byproduct, we formulate a framework that is taking shape in recent work on the Bukh–Conlon conjecture.