Minimal completely separating systems

Minimal completely separating systems of sets

Let [n] denote an n-set. A subset,) of [n] i from j if i and j t/:. 5'. A collection of k-sets n called (n, k) completely if, for each ordered pair j) E [n] x [71,] with i n which i from j. Let denote the size of a smallest (71" k) completely Amongst other things, it will be shown that R( 71" k) for n > except when n (Hl) 1, and R(71" k) k + 1 for G) n k 2 /2. These results build on and extend ...

Minimal (n) and (n, h, k) Completely Separating Systems

Completely separating systems of k-sets

Dickson On a problem concerning separating systems of a nite set, Journal of Combinatorial Theory, 7 (1969), 191{196.] introduced the notion of a completely separating set system. We study such systems with the additional constraint that each set in the system has the same size. Let T denote an n-set. We say that a subset S of T separates i from j if i 2 S and j 6 2 S. A collection of k-sets C ...

On the number of minimal completely separating systems and antichains in a Boolean lattice

An (n)completely separating system C ((n)CSS) is a collection of blocks of [n] = {1, . . . , n} such that for all distinct a, b ∈ [n] there are blocks A,B ∈ C with a ∈ A \B and b ∈ B \A. An (n)CSS is minimal if it contains the minimum possible number of blocks for a CSS on [n]. The number of non-isomorphic minimal (n)CSSs is determined for 11 ≤ n ≤ 35. This also provides an enumeration of a nat...

Completely separating systems of k-sets for 7 ≤ k ≤ 10