On the Turán number of 1-subdivision of $K_{3,t}$

Abstract

For a graph H, the 1-subdivision of H, denoted by H ′ , is the graph obtained by replacing the edges of H by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For any integer s ≥ 2, estimate the smallest t such that ex(n, K ′ s,t ) = Ω(n). In this paper, we consider the case s = 3. More precisely, we provide an explicit construction giving ex(n, K ′ 3,30 ) = Ω(n3 ), which reduces the estimation for the smallest value of t from a magnitude of 10 56 to the number 30. The construction is algebraic, which is based on some equations over finite fields.