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Title: The Zarankiewicz problem in 3‐partite graphs

Letbe a graph,be an integer, and writefor the maximum number of edges in an‐vertex graph that is‐partite and has no subgraph isomorphic to. The functionhas been studied by many researchers. Findingis a special case of the Zarankiewicz problem. We prove an analog of the Kövári‐Sós‐Turán theorem for 3‐partite graphs by showingurn:x-wiley:10638539:media:jcd21654:jcd21654-math-0009for. Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case thatandis odd, that is,for. In the cases ofand, we use a result of Allen, Keevash, Sudakov, and Verstraëte, to show that a similar upper bound holds for alland gives a better constant when. Finally, we point out an interesting connection between difference families from design theory and.

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Author(s) / Creator(s):
Publisher / Repository:
Wiley Blackwell (John Wiley & Sons)
Date Published:
Journal Name:
Journal of Combinatorial Designs
Page Range / eLocation ID:
p. 391-405
Medium: X
Sponsoring Org:
National Science Foundation
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