OFFSET
1,2
COMMENTS
A collection has Property B if the elements of the sets in C can be partitioned into sets such that every set in C contains at least 1 element from each set.
Given a finite set X, a collection C of subsets of X has Property B if X can be partitioned into two disjoint subsets Y and Z such that every set in C meets both Y and Z.
Any hypergraph whose vertices can be 2-colored such that every hyperedge contains at least one vertex of each of the 2 colors has Property B.
29 <= a(5) <= 51, last improved in 2020 (from a lower bound of 28 to 29).
LINKS
Sachin Aglave, V. A. Amarnath, Saswata Shannigrahi, and Shwetank Singh, Improved bounds for uniform hypergraphs without property B, Australasian Journal of Combinatorics, 76 (1) (2020): 73-86. ISSN 2202-3518.
Patric R. J. Östergård, On the minimum size of 4-uniform hypergraphs without property B, Discrete Applied Mathematics. 163, Part 2 (2014) 199-204.
Wikipedia, Property B.
Kyle Wood, Sets for a(4)=23
EXAMPLE
For a(2)=3 we have the sets {1, 2}, {1, 3}, {2, 3}.
For a(3)=7 we have the sets {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {5, 6, 1}, {6, 7, 2}, {7, 1, 3}.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Kyle Wood, Jan 02 2026
STATUS
approved
