Abstract
For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|≥c t ·n 2−1/t which do not contain a complete bipartite graghK t,t !+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K t,s ) for any fixedt and alls≥t!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.
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Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.
Research supported in part by Hungarian National Foundation for Scientific Research Grant