An extremal problem for sets with applications to graph theory

Noga Alon

doi:10.1016/0097-3165(85)90048-2

An extremal problem for sets with applications to graph theory

Noga Alon^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Let X₁, ..., X_n be n disjoint sets. For 1 ≤ i ≤ n and 1 ≤ j ≤ h let A_ij and B_ij be subsets of X_i that satisfy |A_ij| ≤ r_i and |B_ij| ≤ s_i for 1 ≤ i ≤ n, 1 ≤ j ≤ h, (∪_i A_ij) ∩ (∪_i B_ij) = {circled division slash} for 1 ≤ j ≤ h, (∪_i A_ij) ∩ (∪_i B_il) ≠ {circled division slash} for 1 ≤ j < l ≤ h. We prove that h≤ Π i=1 n r_i+s_i r_i. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).

Original language	English
Pages (from-to)	82-89
Number of pages	8
Journal	Journal of Combinatorial Theory - Series A
Volume	40
Issue number	1
DOIs	https://doi.org/10.1016/0097-3165(85)90048-2
State	Published - Sep 1985
Externally published	Yes

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title = "An extremal problem for sets with applications to graph theory",

abstract = "Let X1, ..., Xn be n disjoint sets. For 1 ≤ i ≤ n and 1 ≤ j ≤ h let Aij and Bij be subsets of Xi that satisfy |Aij| ≤ ri and |Bij| ≤ si for 1 ≤ i ≤ n, 1 ≤ j ≤ h, (∪i Aij) ∩ (∪i Bij) = {circled division slash} for 1 ≤ j ≤ h, (∪i Aij) ∩ (∪i Bil) ≠ {circled division slash} for 1 ≤ j < l ≤ h. We prove that h≤ Π i=1 n ri+si ri. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).",

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AB - Let X1, ..., Xn be n disjoint sets. For 1 ≤ i ≤ n and 1 ≤ j ≤ h let Aij and Bij be subsets of Xi that satisfy |Aij| ≤ ri and |Bij| ≤ si for 1 ≤ i ≤ n, 1 ≤ j ≤ h, (∪i Aij) ∩ (∪i Bij) = {circled division slash} for 1 ≤ j ≤ h, (∪i Aij) ∩ (∪i Bil) ≠ {circled division slash} for 1 ≤ j < l ≤ h. We prove that h≤ Π i=1 n ri+si ri. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).

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An extremal problem for sets with applications to graph theory

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