The de Bruijn-Erdo″s theorem for hypergraphs

Noga Alon, Keith E. Mellinger, Dhruv Mubayi*, Jacques Verstraëte

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Fix integers n ≤ r ≤ 2. A clique partition of ( r [n] ) is a collection of proper subsets A 1, A 2,.. ., A t ⊂ [n] such that ∪ i ( r Ai) is a partition of ( r [n]). Let cp(n, r) denote the minimum size of a clique partition of ( r [n]). A classical theorem of de Bruijn and Erdo″s states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≤ 3, cp(n, r) ≤ (1 + o(1))n r/2 as n → ∞. We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman-Mullin-Stinson. We give further evidence of this conjecture by constructing, for each r ≤ 4, a family of (1+o(1))n r/2 subsets of [n] with the following property: no two r -sets of [n] are covered more than once and all but o(n r) of the r -sets of [n] are covered. We also give an absolute lower bound cp(n, r) ≤ (nr)/( r q+r-1) when n = q 2 + q + r - 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

Original languageEnglish
Pages (from-to)233-245
Number of pages13
JournalDesigns, Codes, and Cryptography
Volume65
Issue number3
DOIs
StatePublished - Dec 2012

Funding

FundersFunder number
USA-Israeli BSF
National Science FoundationDMS-0800704, DMS-0969092, DMS-0653946
Engineering Research Centers

    Keywords

    • De Bruijn-Erdo″s
    • Hypergraph
    • Zarankiewicz problem

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