On Turán exponents of bipartite graphs

Abstract A long-standing conjecture of Erd?s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this is known to be true only rationals form $1+1/k$ $2-1/k$ , integers $k\geq 2$ In paper, we add new which the true: $2-2/(2k+1)$ This in turn also gives an affirmative answer question Pinchasi Sharir on cube-like graphs. Recently, version $^{\prime}$ s conjecture, where one replaces single by finite family, was confirmed Bukh Conlon. They proposed construction graphs should satisfy conjecture. Our result can viewed as first step towards verifying Conlon We prove upper bound Turán theta asymmetric setting employ obtain another exponent exponents: $r=7/5$