Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 233–245

# The de Bruijn–Erdős theorem for hypergraphs

• Noga Alon
• Keith E. Mellinger
• Dhruv Mubayi
• Jacques Verstraëte
Article

## Abstract

Fix integers nr ≥ 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 nq 2 + qr − 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.

## Keywords

Hypergraph de Bruijn-Erdős Zarankiewicz problem

05B05 05C65

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## Authors and Affiliations

• Noga Alon
• 1
• Keith E. Mellinger
• 2
• Dhruv Mubayi
• 3
Email author
• Jacques Verstraëte
• 4
1. 1.Schools of Mathematics and Computer Science, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
2. 2.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA
3. 3.Department of Mathematics, Statistics, and Computer ScienceUniversity of IllinoisChicagoUSA
4. 4.Department of MathematicsUniversity of California, San DiegoLa JollaUSA