TY - JOUR

T1 - Intersecting Systems

AU - Ahlswede, R.

AU - Alon, N.

AU - Erdos, P. L.

AU - Ruszinkó, M.

AU - Székely, L. A.

PY - 1997

Y1 - 1997

N2 - An intersecting system of type (∃, ∀, k, n) is a collection double-struck F sign = {ℱ1,...,ℱm} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ℱi and ℱj of distinct members of double-struck F sign there exists an A ∈ ℱi that intersects every B ∈ ℱj. Let In(∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k ≥ 1, there exists an n0(k) so that In(∃, ∀, k) = (n-1k-1) for all n > n0(k). Here we show that this is true for k ≤ 3, but false for all k ≥ 8. We also prove some related results.

AB - An intersecting system of type (∃, ∀, k, n) is a collection double-struck F sign = {ℱ1,...,ℱm} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ℱi and ℱj of distinct members of double-struck F sign there exists an A ∈ ℱi that intersects every B ∈ ℱj. Let In(∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k ≥ 1, there exists an n0(k) so that In(∃, ∀, k) = (n-1k-1) for all n > n0(k). Here we show that this is true for k ≤ 3, but false for all k ≥ 8. We also prove some related results.

UR - http://www.scopus.com/inward/record.url?scp=0031531738&partnerID=8YFLogxK

U2 - 10.1017/S0963548397003003

DO - 10.1017/S0963548397003003

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AN - SCOPUS:0031531738

SN - 0963-5483

VL - 6

SP - 127

EP - 137

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 2

ER -