Turán numbers for Ks,t-free graphs: Topological obstructions and algebraic constructions
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We show that every hypersurface in ℝs × ℝs contains a large grid, i.e., the set of the form S × T, with S, T ⊂ ℝs. We use this to deduce that the known constructions of extremal K2,2-free and K3,3-free graphs cannot be generalized to a similar construction of Ks,s-free graphs for any s ≥ 4. We also give new constructions of extremal Ks,t-free graphs for large t.
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