Property B

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}} (a triangle). 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.

Erdős (1963) proved that for any collection of fewer than ${\displaystyle 2^{n-1}}$ sets of size n, there exists a 2-coloring in which all set are bichromatic. The proof is simple: Consider a random coloring. The probability that an arbitrary set is monochromatic is ${\displaystyle 2^{-n+1}}$. By a union bound, the probability that there exist a monochromatic set is less than ${\displaystyle 2^{n-1}2^{-n+1}=1}$. Therefore, there exists a good coloring.

Erdős (1964) showed the existence of an n-uniform hypergraph with ${\displaystyle O(2^{n}\cdot n^{2})}$ hyperedges which does not have property B (i.e., does not have a 2-coloring in which all hyperedges are bichromatic), establishing an upper bound.

Schmidt (1963) proved that every collection of at most ${\displaystyle n/(n+4)\cdot 2^{n}}$ sets of size n has property B. Erdős and Lovász conjectured that ${\displaystyle m(n)=\theta (2^{n}\cdot n)}$. Beck in 1978 improved the lower bound to ${\displaystyle m(n)=\Omega (n^{1/3-\epsilon }2^{n})}$, where ${\displaystyle \epsilon }$ is an arbitrary small positive number. In 2000, Radhakrishnan and Srinivasan improved the lower bound to ${\displaystyle m(n)=\Omega (2^{n}\cdot {\sqrt {n/\log n}})}$. They used a clever probabilistic algorithm.