Drisko (1998) proved (essentially) that every family of 2n - 1 matchings of size n in a bipartite graph possesses a partial rainbow matching of size n. In Bark et al. (2017) this was generalized as follows: Any [k+2/k...
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Drisko (1998) proved (essentially) that every family of 2n - 1 matchings of size n in a bipartite graph possesses a partial rainbow matching of size n. In Bark et al. (2017) this was generalized as follows: Any [k+2/k+1 n] - (k+1) matchings of size n in a bipartite graph have a rainbow matching of size n - k. The paper has a twofold aim: (i) to extend these results to matchings of not necessarily equal cardinalities, and (ii) to prove a conjecture of Drisko, on the characterization of those families of 2n - 2 matchings of size n in a bipartite graph that do not possess a rainbow matching of size it Combining the latter with an idea of Alon (2011), we re-prove a characterization of the extreme case in a well-known theorem of Erdos-Ginzburg-Ziv in additive number theory. (C) 2017 Elsevier Ltd. All rights reserved.
Let A subset of Z(d) be a finite set. It is known that N A has a particular size (vertical bar N A vertical bar| = P-A(N) for some P-A(X) is an element of Q[X]) and structure (all of the lattice points in a cone other...
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Let A subset of Z(d) be a finite set. It is known that N A has a particular size (vertical bar N A vertical bar| = P-A(N) for some P-A(X) is an element of Q[X]) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when d = 1, when the convex hull of A is a simplex or when vertical bar A vertical bar = d + 2 Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.
Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group T...
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Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group T, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality k contained in an additive basis of order at most h can be bounded in terms of h and k alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon N. Also, using invariant means, we address a classical problem, initiated by Erdos and Graham and then generalized by Nash and Nathanson both in the case of N, of estimating the maximal order XT (h, k) that a basis of cocardinality k contained in an additive basis of order at most h can have. Among other results, we prove that XT (h, k) = O(h2kC1) for every integer k > 1. This result is new even in the case where k = 1. Besides the maximal order XT (h, k), the typical order ST (h, k) is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in N and in abelian groups.
We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely m...
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We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely many primes p, or groups of the form circle plus (p is an element of I) Z(p(infinity)), where I is an infinite set of primes.
A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two c...
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A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of reversible square' type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities.
We study a process of generating random positive integer weight sequences {W-n} where the gaps between the weights {X-n = W-n - Wn-1} are i.i.d. positive integer-valued random variables. The main result of the paper i...
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We study a process of generating random positive integer weight sequences {W-n} where the gaps between the weights {X-n = W-n - Wn-1} are i.i.d. positive integer-valued random variables. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the weight sequence is almost surely asymptotically m-complete for every m >= 2, i.e. every large enough multiple of the greatest common divisor (gcd) of gap values can be written as a sum of m distinct weights for any fixed m >= 2. Under the weaker assumption of finite 1/2-moment for the gap distribution, we also show the simpler result that, almost surely, the resulting weight sequence is asymptotically complete, i.e. all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights.
Let p be a prime, epsilon > 0 and 0 c( N log N)(1/2), c = c(epsilon) > 0. We use this bound to show that any lambda not equivalent to 0 (mod p) can be represented in the form lambda = n(1)!...n(7)! (mod p), whe...
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Let p be a prime, epsilon > 0 and 0 < L + 1 < L + N < p. We prove that if p(1/2+epsilon) < N < p(1-epsilon), then #{n! (mod p);L + 1 <= n <= L + N} > c( N log N)(1/2), c = c(epsilon) > 0. We use this bound to show that any lambda not equivalent to 0 (mod p) can be represented in the form lambda = n(1)!...n(7)! (mod p), where n(i) = o(p(11/12)). This refines the previously known range for n(i).
Given f(x, y) is an element of Z[x, y] with no common components with x(a) - y(b) and x(a)y(b) - 1, we prove that for p sufficiently large, with C(f) exceptions, the solutions (x, y) is an element of (F) over bar (p) ...
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Given f(x, y) is an element of Z[x, y] with no common components with x(a) - y(b) and x(a)y(b) - 1, we prove that for p sufficiently large, with C(f) exceptions, the solutions (x, y) is an element of (F) over bar (p) x (F) over bar (p) of f(x, y) = 0 satisfy ord(x) + ord(y) > c(log p/ log log p)(1/2), where c is a constant and ord(r) is the order of r in the multiplicative group (F) over bar (p)*. Moreover, for most p < N, N being a large number, we prove that, with C(f) exceptions, ord(x) + ord(y) > p(1/4+epsilon(p)), where epsilon(p) is an arbitrary function tending to 0 when p goes to infinity.
Let eta(i), i = 1 ,..., n be iid Bernoulli random variables. Given a multiset v of n numbers v(1) ,..., v(n), the concentration probability P(1)(v) of v is defined as P(1)(v) := sup(x) P(v(1)eta(1)+ ... v(n)eta(n) = x...
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Let eta(i), i = 1 ,..., n be iid Bernoulli random variables. Given a multiset v of n numbers v(1) ,..., v(n), the concentration probability P(1)(v) of v is defined as P(1)(v) := sup(x) P(v(1)eta(1)+ ... v(n)eta(n) = x). A classical result of Littlewood-Offord and Erdos from the 1940s asserts that if the v(i) are nonzero, then this probability is at most O(n(-1/2)). Since then, many researchers obtained better bounds by assuming various restrictions on v. In this article, we give an asymptotically optimal characterization for all multisets v having large concentration probability. This allow us to strengthen or recover several previous results in a straightforward manner. (C) 2010 Wiley Periodicals, Inc. Random Struct. Alg., 37, 525-539, 2010
The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length k in a given set of size n is considered. It is proved that it is sufficient, in a certain sense, to consi...
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The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length k in a given set of size n is considered. It is proved that it is sufficient, in a certain sense, to consider the interval [1,..., n]. The study continues the work of Komls, Sulyok, and Szemeredi.
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