Combinatorial Nullstellensatz and Turán numbers of complete r-partite r-uniform hypergraphs

In this note we describe how Laso & nacute;'s generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Tur & aacute;n number ex(n,K-s1,& mldr;,sr((r))) of the complete r-partite r-uniform hypergraph K-s1,& mldr;,sr((r)). To illustrate the potential of this method, we give a short and simple explicit construction for the Erd & odblac;s box problem, showing that ex(n,K-2,& mldr;,2((r))) = Omega(n(r-1/r)), which asymptotically matches best known bounds when r <= 4. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).