On the rational Turán exponents conjecture

Abstract

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n,F)=Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 0,1,75,2, and the numbers of the form 1+1m, 2−1m, 2−2m for integers m≥1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than the two numbers 1 and 2.In this paper, we make progress on the conjecture of Erdős and Simonovits. First, we show that 2−ab is realisable for any integers a,b≥1 with b>a and b≡±1(moda). This includes all previously known ones, and gives infinitely many limit points 2−1m in the set of all realisable numbers as a consequence.Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.