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A new extremal property of Steiner triple-systems. (English) Zbl 0553.05019
A family $${\mathcal F}$$ of 3-subsets of an n-element set is union-free if, for any 4 subsets F,G,F’,G’$$\in {\mathcal F}$$, $$F\cup G=F'\cup G'$$ implies that $$\{F,G\}=\{F',G'\}.$$ $${\mathcal F}$$ is weakly union-free if, for any 4 distinct subsets F,G,F’,G’$$\in {\mathcal F}$$, $$F\cup G\neq F'\cup G'.$$ Denote by $$f_ 3(n)$$ (respectively $$F_ 3(n))$$ the maximum number of sets in a union-free (respectively weakly union-free) family. Theorem 1.4: $$f_ 3(n)=[n(n-1)/6].$$ (If $$n\equiv 1$$ or $$n\equiv 3$$ (mod 6), $$n\geq 7$$, Steiner triple systems realize the equality of Theorem 1.4; other examples exist.) Theorem 1.6: $$F_ 3(n)\leq n(n-1)/3,$$ and any family for which equality holds is a Steiner triple system. Moreover, if $$n\equiv 1$$ (mod 6), then equality holds for n sufficiently large. Corollary to Theorem 1.6: For n sufficiently large, $$n(n-1)/3-(10/3)n<F_ 3(n)\leq [n(n-1)/3].$$
Reviewer: W.G.Brown

##### MSC:
 05B07 Triple systems 05A05 Permutations, words, matrices 03E05 Other combinatorial set theory
##### Keywords:
family of 3-subsets; union-free
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