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In this note we describe how Lasoń's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Turán number ex(n,Ks1,…,sr(r)) of the complete r-partite r-uniform hypergraph Ks1,…,sr(r). To illustrate the potential of this method, we give a short and simple explicit construction for the Erdős box problem, showing that ex(n,K2,…,2(r))=Ω(nr−1/r), which asymptotically matches best known bounds when r≤4.
Alexey Gordeev (Fri,) studied this question.