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×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×P²) is (2,2)-grid-free, then there exists F∈C[x¯,y¯] of degree ≤2 in y¯ such that H∩(X×P²)={F=0}∩(X×P²). Finally, we transfer the result to algebraically closed fields of large characteristic.