A392185 - OEIS

%I #31 Jan 27 2026 09:59:47

%S 1,3,7,23

%N Smallest number of sets a(n) in a collection C of sets of size n such that C does not have Property B (see comments).

%C 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.

%C 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.

%C 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.

%C 29 <= a(5) <= 51, last improved in 2020 (from a lower bound of 28 to 29).

%H Sachin Aglave, V. A. Amarnath, Saswata Shannigrahi, and Shwetank Singh, <a href="https://ajc.maths.uq.edu.au/pdf/76/ajc_v76_p073.pdf">Improved bounds for uniform hypergraphs without property B</a>, Australasian Journal of Combinatorics, 76 (1) (2020): 73-86. ISSN 2202-3518.

%H Patric R. J. Östergård, <a href="https://doi.org/10.1016/j.dam.2011.11.035">On the minimum size of 4-uniform hypergraphs without property B</a>, Discrete Applied Mathematics. 163, Part 2 (2014) 199-204.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Property_B">Property B</a>.

%H Kyle Wood, <a href="/A392185/a392185_2.txt">Sets for a(4)=23</a>

%e For a(2)=3 we have the sets {1, 2}, {1, 3}, {2, 3}.

%e 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}.

%K nonn,hard,more

%O 1,2

%A _Kyle Wood_, Jan 02 2026