127 research outputs found
Strong Qualitative Independence
AbstractThe subsets A,B of the n-element X are said to be s-strongly separating if the two sets divide X into four sets of size at least s. The maximum number h(n,s) of pairwise s-strongly separating subsets was asymptotically determined by Frankl (Ars Combin. 1 (1976) 53) for fixed s and large n. A new proof is given. Also, estimates for h(n,cn) are found where c is a small constant
Intersecting P-free families
We study the problem of determining the size of the largest intersecting P-free family for a given partially ordered set (poset) P. In particular, we find the exact size of the largest intersecting B-free family where B is the butterfly poset and classify the cases of equality. The proof uses a new generalization of the partition method of Griggs, Li and Lu. We also prove generalizations of two well-known inequalities of Bollobás and Greene, Katona and Kleitman in this case. Furthermore, we obtain a general bound on the size of the largest intersecting P-free family, which is sharp for an infinite class of posets originally considered by Burcsi and Nagy, when n is odd. Finally, we give a new proof of the bound on the maximum size of an intersecting k-Sperner family and determine the cases of equality. © 2017 Elsevier Inc
On Structural Resource of Monotone Recognition
Algorithmic resources are considered for elaboration and identification of monotone functions and
some alternate structures are brought, which are more explicit in sense of structure and quantities and which can
serve as elements of practical identification algorithms. General monotone recognition is considered on multi-
dimensional grid structure. Particular reconstructing problem is reduced to the monotone recognition through the
multi-dimensional grid partitioning into the set of binary cubes
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 natural class of antichains
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