NDLI: The de Bruijn–Erdős theorem for hypergraphs

Content Provider	Springer Nature Link
Author	Alon, ga Mellinger, Keith E. Verstraëte, Jacques Mubayi, Dhruv
Copyright Year	2011
Abstract	Fix integers n ≥ r ≥ 2. A clique partition of $${{[n] \choose r}}$$ is a collection of proper subsets $${A_1, A_2, \ldots, A_t \subset [n]}$$ such that $${\bigcup_i{A_i \choose r}}$$ is a partition of $${{[n]\choose r}}$$ . Let cp(n, r) denote the minimum size of a clique partition of $${{[n] \choose r}}$$ . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, $${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$ 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 $${{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}$$ 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.
Ending Page	245
Page Count	13
Starting Page	233
File Format	PDF
ISSN	09251022
e-ISSN	15737586
Journal	Designs, Codes and Cryptography
Issue Number	3
Volume Number	65
Language	English
Publisher	Springer US
Publisher Date	2011-08-27
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hypergraph de Bruijn-Erdős Information and Communication, Circuits Data Structures, Cryptology and Information Theory Discrete Mathematics in Computer Science Data Encryption Coding and Information Theory Block designs Zarankiewicz problem Hypergraphs Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science Applications

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