Author:
Frankl Peter,Pach Janos,Pálvölgyi Dömötör
Abstract
Extending the notion of sunflowers, we call a family of at least two sets an \emph{odd-sunflower} if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi conjecture, recently proved by %Alweiss, Lovett, Wu, and Zhang, Naslund and Sawin, that there is a constant $\mu <2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most $\mu^n$ sets. We construct such families of size at least $1.5021^n$.
Reference18 articles.
1. H. L. Abbott, D. Hanson: On finite Δ-systems II, Discrete Math., 17 (1977), 121--126.
2. H. L. Abbott, D. Hanson, and N. Sauer: Intersection theorems for systems of sets, J. Combin. Theory Ser. A, 12 (3) (1972), 381--389.
3. N. Alon and R. Holzman: Near-sunflowers and focal families, preprint, https://arxiv.org/abs/2010.05992.
4. R. Alweiss, S. Lovett, K. Wu, and J. Zhang: Improved bounds for the sunflower lemma, Ann. of Math. (2) 194 (2021), no. 3, 795--815.
5. L. Babai and P. Frankl: Linear Algebra Methods in Combinatorics (Preliminary Version 2, Dept. of Computer Science, The University of Chicago, 1992.