Property B

In mathematics, Property B is a certain set theoretic property. Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z. The smallest number of sets in a collection of sets of size n such that C does not have Property B is denoted by m(n).

The property gets its name from mathematician Felix Bernstein, who first introduced the property in 1908.

Property B is equivalent to 2-coloring the hypergraph described by the collection C. A hypergraph with property B is also called bipartite, by analogy to the bipartite graphs.

Values of m(n)

It is known that m(1) = 1, m(2) = 3, and m(3) = 7 (as can by seen by the following examples); the value of m(4) = 23 (Östergård), although finding this result was the result of an exhaustive search. An upper bound of 23 (Seymour, Toft) and a lower bound of 21 (Manning) have been proven. At the time of this writing (March 2017), there is no OEIS entry for the sequence m(n) yet, due to the lack of terms known.

m(1): For n = 1, set X = {1}, and C = {{1}}. Then C does not have Property B.

m(2): For n = 2, set X = {1, 2, 3} and C = {{1, 2}, {1, 3}, {2, 3}}. Then C does not have Property B, so m(2) <= 3. However, C' = {{1, 2}, {1, 3}} does (set Y = {1} and Z = {2, 3}), so m(2) >= 3.

m(3): For n = 3, set X = {1, 2, 3, 4, 5, 6, 7}, and C = {{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {5, 6, 1}, {6, 7, 2}, {7, 1, 3}} (the Steiner triple system S₇); C does not have Property B (so m(3) <= 7), but if any element of C is omitted, then that element can be taken as Y, and the set of remaining elements C' will have Property B (so for this particular case, m(3) >= 7). One may check all other collections of 6 3-sets to see that all have Property B.

m(4): Östergård (2014) through an exhaustive search found m(4) = 23. Seymour (1974) constructed a hypergraph on 11 vertices with 23 edges without Property B, which shows that m(4) <= 23. Manning (1995) narrowed the floor such that m(4) >= 21.

Asymptotics of m(n)

Erdős (1963) proved that for any collection of fewer than $2^{n-1}$ sets of size n, there exists a 2-coloring in which no set is monochromatic. The proof is simple: Consider a random coloring. The probability that an arbitrary set is monochromatic is $2^{-n+1}$ . By a union bound, the probability that there exist a monochromatic set is less than $2^{n-1}2^{-n+1}=1$ . Therefore, there exists a good coloring.

Erdős (1964) showed the existence of an n-uniform hypergraph with $O(2^{n}\cdot n^{2})$ hyperedges which does not have property B, establishing an upper bound. Schmidt (1963) proved that every collection of at most $n/(n+4)\cdot 2^{n}$ sets of size n has property B. Erdős and Lovász conjectured that $m(n)=\theta (2^{n}\cdot n)$ . Beck in 1978 improved the lower bound to $m(n)=\Omega (n^{1/3-\epsilon }2^{n})$ , where $\epsilon$ is an arbitrary small positive number. In 2000, Radhakrishnan and Srinivasan improved the lower bound to $m(n)=\Omega (2^{n}\cdot {\sqrt {n/\log n}})$ . They used a clever probabilistic algorithm.

References

Bernstein, F. (1908), "Zur theorie der trigonometrische Reihen", Leipz. Ber., 60: 325–328.
Erdős, Paul (1963), "On a combinatorial problem", Nordisk Mat. Tidskr., 11: 5–10
Erdős, P. (1964). "On a combinatorial problem. II". Acta Mathematica Academiae Scientiarum Hungaricae. 15 (3–4): 445–447. doi:10.1007/BF01897152.
Schmidt, W. M. (1964). "Ein kombinatorisches Problem von P. Erdős und A. Hajnal". Acta Mathematica Academiae Scientiarum Hungaricae. 15 (3–4): 373–374. doi:10.1007/BF01897145.
Seymour, Paul (1974), "A note on a combinatorial problem of Erdős and Hajnal", Bulletin of the London Mathematical Society, 8: 681–682, doi:10.1112/jlms/s2-8.4.681.
Toft, Bjarne (1975), "On colour-critical hypergraphs", in Hajnal, A.; Rado, Richard; Sós, Vera T. (eds.), Infinite and Finite Sets: To Paul Erdös on His 60th Birthday, North Holland Publishing Co., pp. 1445–1457.
Manning, G. M. (1995), "Some results on the m(4) problem of Erdős and Hajnal", Electronic Research Announcements of the American Mathematical Society, 1: 112–113, doi:10.1090/S1079-6762-95-03004-6.
Beck, J. (1978), "On 3-chromatic hypergraphs", Discrete Mathematics, 24 (2): 127–137, doi:10.1016/0012-365X(78)90191-7.
Radhakrishnan, J.; Srinivasan, A. (2000), "Improved bounds and algorithms for hypergraph 2-coloring", Random Structures and Algorithms, 16 (1): 4–32, doi:10.1002/(SICI)1098-2418(200001)16:1<4::AID-RSA2>3.0.CO;2-2.
Miller, E. W. (1937), "On a property of families of sets", Comp. Rend. Varsovie, 30: 31–38.
Erdős, P.; Hajnal, A. (1961), "On a property of families of sets", Acta Math. Acad. Sci. Hung., 12: 87–123, doi:10.1007/BF02066676.
Abbott, H. L.; Hanson, D. (1969), "On a combinatorial problem of Erdös", Canadian Mathematical Bulletin, 12: 823–829, doi:10.4153/CMB-1969-107-x
Östergård, Patric R. J. (30 January 2014). "On the minimum size of 4-uniform hypergraphs without property". Discrete Applied Mathematics. 163, Part 2: 199–204. doi:10.1016/j.dam.2011.11.035. Retrieved 7 March 2017.

.