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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  and of  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 ∈  , 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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