Turán numbers for Ks,t-free graphs: Topological obstructions and algebraic constructions

We show that every hypersurface in ℝ s × ℝ s contains a large grid, i.e., the set of the form S × T , with S, T ⊂ ℝ s . We use this to deduce that the known constructions of extremal K 2,2 -free and K 3,3 -free graphs cannot be generalized to a similar construction of K s,s -free graphs for any s ≥ 4. We also give new constructions of extremal K s,t -free graphs for large t .