Many Turan exponents via subdivisions
arXiv:1908.02385
The paper proves that for any positive integers p and q with q > p², the exponent 1 + p/q can be realized as a Turán exponent of some bipartite graph, expanding the known family of Turán exponents via graph subdivisions.
Abstract
Given a graph $H$ and a positive integer $n$, the {\it Turán number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Turán exponent} if there exists a bipartite graph $H$ such that $\ex(n,H)=Î(n^r)$. A long-standing conjecture of ErdÅs and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q> p$. In this paper, we build on recent developments on the conjecture to establish a large family of new Turán exponents. In particular, it follows from our main result that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q> p^2$.
20 pages