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9085 results in Discrete Mathematics, Information Theory and Coding

Page 85 of 455

  • Multicolour Sunflowers

  • Part of
  • DHRUV MUBAYI, LUJIA WANG
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 6 / November 2018
    Published online by Cambridge University Press:
    22 April 2018, pp. 974-987

  • 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.

Page 85 of 455