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### Sum-Free Sets and Related Sets

Yuri Bilu
1998 Combinatorica
A set A of integers is sum-free if A\(A+A) = ;. Cameron conjectured that the number of sum-free sets A f1; : : : ; Ng is O(2 N=2 ).  ...  As a step towards this conjecture, we prove that the number of sets A f1; : : : ; Ng satisfying (A + A + A) \ (A + A + A + A) = ; is 2 (N+1)=2] (1 + o(1)).  ...  2 =16)N sum-free sets A f1; : : : ; Ng such that and Erd} os 4] proved that there are at most O 2 N=2 sum-free subsets of f N=3]; : : : ; Ng.  ...

### Parametric matroid of rough set [article]

Yanfang Liu, William Zhu
2012 arXiv   pre-print
Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid.  ...  On the one hand, for a universe and an equivalence relation on the universe, a parametric set family is defined through the lower approximation operator.  ...  For a universe and an equivalence relation on the universe, the parametric matroid of the rough set with respect to a subset of the universe is the direct sum of a partitioncircuit matroid and a free matroid  ...

### Parametric Matroid of Rough Set

Yanfang Liu, Hong Zhao, William Zhu
2015 International Journal of Uncertainty Fuzziness and Knowledge-Based Systems
Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid.  ...  Let R be an equivalence relation on U and X ⊆ U .  ...  For a universe and an equivalence relation on the universe, the parametric matroid of the rough set with respect to a subset of the universe is the direct sum of a partitioncircuit matroid and a free matroid  ...

### On the diagonal queens domination problem

E.J Cockayne, S.T Hedetniemi
1986 Journal of combinatorial theory. Series A
K is a diagonal dominating set if and only if N -K is a midpoint-free, even-sum set. ,, 2a, ,..., 2a,} or (2a, -1, 2a, -l,..., 2ak -l} is a midpoint-free, even-sum set of ( l,..., n > if and only if  ...  The following elementary result simplifies the computation of midpointfree, even-sum sets and enables us to relate diag(n) to a well-studied, number-theoretic function. We omit the proof.  ...

### Page 895 of Mathematical Reviews Vol. , Issue 84c [page]

1984 Mathematical Reviews
Centrum, Amsterdam, 1971; MR 49 #4778] that if « is an infinite cardinal <2, and |f(x)|<« for every x, then a free set of cardinality 2*° exists.  ...  Two new operations on fuzzy sets, “drastic sumand “drastic product”, are introduced and their properties and combinations with other operations are exhaustively classified under all such headings as  ...

### Zero-sum free sets with small sum-set

Gautami Bhowmik, Immanuel Halupczok, Jan-Christoph Schlage-Puchta
2011 Mathematics of Computation
Let A be a zero-sum free subset of Z n with |A| = k. We compute for k ≤ 7 the least possible size of the set of all subset-sums of A.  ...  If there are no zero-sum free sets of cardinality k in Z n , we set f n (k) = ∞.  ...  To check that the inequalities are satisfied, too, it suffices to verify that the resulting set B = {x 1 , . . . , x 5 } indeed consists of k different elements and is zero-sum free and that the sum-set  ...

### Page 3727 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews
Summary: “Cameron introduced a natural probability measure on the set Y of sum-free sets, and asked which sets of sum-free sets have a positive probability of occurring in this probability measure.  ...  He showed that the set of subsets of the odd numbers has a positive probability, and that the set of subsets of any sum- free set corresponding to a complete modular sum-free set also has a positive probability  ...

### Zero-sum free sequences with small sum-set [article]

Gautami Bhowmik, Jan-Christoph Schlage-Puchta
2008 arXiv   pre-print
Let A be a zero-sum free subset of Z_n with |A|=k. We compute for k\le 7 the least possible size of the set of all subset-sums of A.  ...  If there are no zero-sum free sets of cardinality k in Z n , we set f n (k) = ∞.  ...  Define f n (k) = min |Σ(B)| − 1, where B runs over all zero-sum free subsets of Z n of cardinality k, and set f (k) := min n f n (k).  ...

### On Recognising Certain One Relation Presentations

James McCool, Alfred Pietrowski
1972 Proceedings of the American Mathematical Society
It follows from the Lemma that Hi is a one relation presentation if and only if {b2, bxbibxl, • • ■ , ¿Î_1è2AÎ_i} is a free generating set of Nx.  ...  This will be the case if and only if {0¿>2, (¡(b^bî1), • • • , 0(¿>Í-1A26Í-Í)} is a free generating set of N2.  ...

### On recognising certain one relation presentations

James McCool, Alfred Pietrowski
1972 Proceedings of the American Mathematical Society
It follows from the Lemma that Hi is a one relation presentation if and only if {b2, bxbibxl, • • ■ , ¿Î_1è2AÎ_i} is a free generating set of Nx.  ...  This will be the case if and only if {0¿>2, (¡(b^bî1), • • • , 0(¿>Í-1A26Í-Í)} is a free generating set of N2.  ...

### Logic Without Syntax [article]

Dominic Hughes
2005 arXiv   pre-print
Abstract propositions correspond to objects of the category G(Rel^L) where G is the Hyland-Tan double glueing construction, Rel is the standard category of sets and relations, and L is a set of literals  ...  We prove that the free binary product-sum category (contraction-weakening logic) over L is a full subcategory of Gl(Rel^L), and the free distributive lattice category (contraction-weakening-distribution  ...  THEOREM 1 CW K is the free binary product-sum category generated by the set K. Proof.  ...

### Theoretical Bounds on the Size of Condensed Representations [chapter]

Nele Dexters, Toon Calders
2005 Lecture Notes in Computer Science
More concrete, we compute a lower bound for the size of the database in terms of the size of the l-free set, and when the database size is smaller than this lower bound, we know that the set cannot be  ...  We will bound the maximal cardinality of an l-free set based on the size of the database.  ...  In this way, we extend results of [7] and of [10] that relate the database size to the maximal length of respectively the non-derivable itemsets and the generalized disjunction free sets.  ...

### On Infinite Sum-free Sets of Natural Numbers

Tomasz łuczak, Tomasz Schoen
1997 Journal of Number Theory
It is shown that every k-sum-free set with upper density larger than 1Â(k+1) is a subset of a periodic k-sum-free set and that each k-sum-free set with upper density larger than 2Â(k+3) is subset of a  ...  In particular, no k-sum-free set has upper density larger than 1Â\ 1 (k), where \ 1 (k)=min[i: i |% k&1], as conjectured by Calkin and Erdo s.  ...  Furthermore, either of relations l (a+ g)+a l+1 + } } } +a k =a+ g and a 1 + } } } +a k =a+ g implies that a # kA which contradicts the fact that a belongs to the k-sum-free set A.  ...

### Maximum number of sum-free colorings in finite abelian groups [article]

Hiep Hàn, Andrea Jiménez
2017 arXiv   pre-print
On the contrary, if the largest sum-free set in G is not unique then A attains κ_r,G if and only if it is the union of two largest sum-free sets (in case r=4) and the union of three ("independent") largest  ...  sum-free sets (in case r=5).  ...  Consider the canonical correspondence relating B i , i ∈ [3] , to the set {i}. Then there are four further largest sum-free sets contained in B 1 ∪B 2 ∪B 3 .  ...

### Page 1686 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews
/4) maximum sum-free sets. Theorem 2: The size of the smallest maximum sum- free set is between c),/n and c2,/n.  ...  2000c:05144 05 05D Extremal combinatorics 2000c:05144 05D05 11B75 Cameron, Peter J. (4-LNDQM; London); Erdés, Paul [Erdés, Paul'] (H-AOS; Budapest) Notes on sum-free and related sets.  ...
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