Bipartite algebraic graphs without quadrilaterals

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 x 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) over bar similar to (v) over bar if and only if ((u) over bar, (v) over bar) is an element of H. Motivated by the Turan problem for bipartite graphs, we say that H boolean AND(X 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) over bar is bounded by a constant d = d(s, t), and we discuss possible notions of the equivalence. We establish the result that if H boolean AND(X x P-2) is (2, 2)-grid-free, then there exists F is an element of CR, [(x) over bar, (y) over bar] of degree < 2 in y such that H boolean AND (X x P-2) = {F = 0} boolean AND (X x P-2). Finally, we transfer the result to algebraically closed fields of large characteristic. (C) 2018 Elsevier B.V. All rights reserved.