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 iff 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 of size n that do 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.

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) is not known, although an upper bound of 23 (Seymour, Toft) and a lower bound of 21 (Manning) have been proven. At the time of this writing (August 2004), 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.

Asymptotics of m(n)

Erdős 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 any one set is monochromatic is $2^{-n+1}$. By a union bound, the probability that any set is monochromatic is less than $2^{n-1}2^{-n+1}=1$. Therefore, there exists a good coloring.

Erdős constructed an n-uniform graph with $O(2^{n}\cdot n^{2})$ edges which does not have property B, establishing an upper bound. Erdős and Lovász conjectured that $m(n)=\theta (2^{n}\cdot n)$. Beck in 1978 improved the lower bound to $m(n)=\Omega (n^{1/3}2^{n})$. 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. Zur theorie der trigonometrische Reihen. Leipz. Ber. 60 (1908): 325-328.
• Seymour, A note on a combinatorial problem of Erdős and Hajnal, Bull. London Math. Soc. 2:8 (1974), 681-682
• Toft, On colour-critical hypergraphs, in Infinite and Finite Sets, ed. A. Hajnal et al, North Holland Publishing Co., 1975, 1445–1457
• G. M. Manning, Some results on the m(4) problem of Erdős and Hajnal, Electron. Research Announcements of the American Mathematical Society, 1(1995) 112-113
• J. Beck, On 3-chromatic hypergraphs. Discrete Math., 24(2):127-137, 1978.
• J. Radhakrishnan and A. Srinivasan, [Improved bounds and algorithms for hypergraph 2-coloring. http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00743519 ] Random Structures and Algorithms, 16(1):4-32, 2000.
• E. W. Miller, ``On a property of families of sets, Comp. Rend. Varsovie (1937), 31-38.
• P. Erdös and A. Hajnal, ``On a property of families of sets, Acta Math. Acad. Sci. Hung., 12 (1961), 87-123.
• H. L. Abbott and D. Hanson, ``On a çombinatorial problem of Erdös, Canad. Math. Bull., 12 (1969), 823-829.