Turلn numbers for Ks,t-free graphs: topological obstructions and algebraic constructions

We show that the constructions of extremal Ktsub2,2-free and K3,3-free graphs cannot be generalized to a similar construction of Ks,s-free graphs for s ⩾ 4 . We also give new constructions of extremal Ks,t-free graphs for large t.