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Multicolour Sunflowers

Part of: Extremal combinatorics

Published online by Cambridge University Press:  22 April 2018

DHRUV MUBAYI  and
LUJIA WANG
DHRUV MUBAYI
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: mubayi@uic.edu, lwang203@uic.edu)
LUJIA WANG
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: mubayi@uic.edu, lwang203@uic.edu)
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Abstract

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A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection C of all of them, and |C| is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi (solved recently in [7, 9]; see also [22]) was that the maximum size of a family of subsets of [n] that contains no sunflower of fixed size k > 2 is exponentially smaller than 2n as n → ∞. We consider the problems of determining the maximum sum and product of k families of subsets of [n] that contain no sunflower of size k with one set from each family. For the sum, we prove that the maximum is

$$(k-1)2^n+1+\sum_{s=0}^{k-2}\binom{n}{s}$$
for all n ⩾ k ⩾ 3, and for the k = 3 case of the product, we prove that the maximum is
$$\biggl(\ffrac{1}{8}+o(1)\biggr)2^{3n}.$$
We conjecture that for all fixed k ⩾ 3, the maximum product is (1/8+o(1))2kn.

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05D40: Probabilistic methods

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 6 , November 2018 , pp. 974 - 987
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†

Research partially supported by NSF grants DMS-0969092 and DMS-1300138.

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