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25994 results in Abstract analysis

Page 126 of 1300

  • On Turán exponents of bipartite graphs

  • Part of
  • Tao Jiang, Jie Ma, Liana Yepremyan
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 2 / March 2022
    Published online by Cambridge University Press:
    04 August 2021, pp. 333-344

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

Page 126 of 1300