58,487 research outputs found
Union-intersecting set systems
Three intersection theorems are proved. First, we determine the size of the
largest set system, where the system of the pairwise unions is l-intersecting.
Then we investigate set systems where the union of any s sets intersect the
union of any t sets. The maximal size of such a set system is determined
exactly if s+t4. Finally, we exactly determine
the maximal size of a k-uniform set system that has the above described
(s,t)-union-intersecting property, for large enough n.Comment: 9 page
Some combinatorial theorems with an application to a problem in number theory
The main object of this thesis is to study the following extremal problem in number theory: Let n and k be integers satisfying n ≥ k ≥ 3. Denote by f(n,k) the largest positive integer for which there exists a set S of f(n,k) integers satisfying -- (i) S ⊑ { 1,2...,n } and -- (ii) no k numbers in S have pairwise the same greatest common divisor. -- We investigate the behaviour of f(n,k) in the case where k → ∞ with n. In particular we obtain estimates for f(n, [logαn]) for fixed α > 0 and f(n,[nα]) for fixed α, 0 < α < 1. -- In the course of our investigations we make use of certain intersection theorems for systems of finite sets. We also include a number of new results concerning these theorems
Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices
A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most sets if it is uniform and at most sets if it is nonuniform. They also prove modular versions of these results.
We consider the following extension of these problems. Call the set systems {\em L-cross-intersecting} if for every pair of distinct sets A,B with and for some the intersection size lies in . For any k and for n > n 0 (s) we give tight bounds on the maximum of . It is at most if the systems are uniform and at most if they are nonuniform. We also obtain modular versions of these results.
Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system on [n] is a matrix M with rows indexed by and columns by the subsets of [n] of size at most s, where if and with , we define M AB to be 1 if and 0 otherwise. Our bound generalizes the well-known result that if , then M has full rank . In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Strongly intersecting integer partitions
We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
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