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Union-free families of sets and equations over fields. (English) Zbl 0589.05013
A collection $${\mathcal F}$$ of k-element subsets of an n-element set is said to be union-free if the sets $$F_ 1\cup F_ 2(F_ 1,F_ 2\in {\mathcal F})$$ are all distinct. Let $$F_ k(n)$$ denote the size of the largest union-free $${\mathcal F}$$. It is shown that $$F_ k(n)$$ lies between two positive multiples of $$n^{\lceil 4k/3\rceil}$$. The proof of the lower bound involves the use of elementary symmetric polynomials. A similar result is obtained for weakly union-free collections, where, for any four distinct sets A,B,A’,B’$$\in {\mathcal F}$$, $$A\cup B=A'\cup B'$$ implies $$\{A,B\}=\{A',B'\}$$. A recent paper of the authors [Discrete Math. 48, 205-212 (1984; Zbl 0553.05019)] relates the case $$k=3$$ to Steiner triple systems.
Reviewer: I.Anderson

##### MSC:
 05A99 Enumerative combinatorics 05A05 Permutations, words, matrices 11B99 Sequences and sets
##### Keywords:
extremal set theory; subsets; Steiner triple systems
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##### References:
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