Bipartite algebraic graphs without quadrilaterals

Abstract

Let P s be the s-dimensional complex projective space, and let X, Y be two non-empty open subsets of P s in the Zariski topology. A hypersurface H in P s × P s induces a bipartite graph G as follows: the partite sets of G are X and Y , and the edge set is defined by u ∼ v if and only if (u, v) ∈ H. Motivated by the Turán problem for bipartite graphs, we say that H ∩ (X × Y ) is (s, t)-grid-free provided that G contains no complete bipartite subgraph that has s vertices in X and t vertices in Y . We conjecture that every (s, t)-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in y is bounded by a constant d = d(s, t), and we discuss possible notions of the equivalence.We establish the result that if H ∩ (X × P 2 ) is (2, 2)-grid-free, then there exists F ∈ C[x, y] of degree ≤ 2 in y such that H ∩ (X × P 2 ) = {F = 0} ∩ (X × P 2 ). Finally, we transfer the result to algebraically closed fields of large characteristic. t−1 n 2 + o(n 2 ). When F is not bipartite, this gives an asymptotic result for the Turán number. On the other hand, for all but few bipartite graphs F , the order of ex(n, F ) is not known. Most of the research on this problem focused on two classes of graphs: complete bipartite graphs and cycles of even length. A comprehensive survey is given by Füredi and Simonovits [FS13].

Random polynomial graphs for random Turán problems

“…Only sporadic results for $$\text{ex}(n,{\rm{ {\mathcal F} }})$$ are known when $${\rm{ {\mathcal F} }}$$ contains a bipartite graph, and in most cases these bounds are not tight. For example, the Kővári–Sós–Turán theorem [30] implies $$\text{ex}(n,{K}_{s,t})=O({n}^{2-1\unicode{x02215}s})$$, and this bound is only known to be tight when $$t$$ is sufficiently large in terms of $$s$$; see, for example, recent work of Bukh [6].…”