### 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 language | English (US) |
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Pages (from-to) | 233-245 |

Number of pages | 13 |

Journal | Designs, Codes, and Cryptography |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2012 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Applied Mathematics

### Keywords

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

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## Cite this

*Designs, Codes, and Cryptography*,

*65*(3), 233-245. https://doi.org/10.1007/s10623-011-9555-4