scholarly journals Groups Containing Small Locally Maximal Product-Free Sets

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Chimere S. Anabanti ◽  
Sarah B. Hart
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Locally Maximal ◽  
Free Set ◽  
Free Sets

Let G be a group and S a nonempty subset of G. Then, S is product-free if ab∉S for all a,b∈S. We say S is a locally maximal product-free set if S is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in G. Alternatively, given a positive integer k, one can ask the following: what is the largest integer nk such that there is a group of order nk with a locally maximal product-free set of size k? The groups containing locally maximal product-free sets of sizes 1 and 2 are known, and it has been conjectured that n3=24. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence nk.

scholarly journals Groups containing locally maximal product-free sets of size 4

2021 ◽  
Vol 31 (2) ◽  
pp. 167-194
Author(s):  
C. S. Anabanti ◽  
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Open Problem ◽  
Finite Abelian Group ◽  
Locally Maximal ◽  
A Finite Group ◽  
Free Set ◽  
Free Sets

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

scholarly journals Maximum $k$-Sum $\mathbf{n}$-Free Sets of the 2-Dimensional Integer Lattice

10.37236/8895 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ilkyoo Choi ◽  
Ringi Kim ◽  
Boram Park
Keyword(s):  
Positive Integer ◽  
Lattice Point ◽  
Higher Dimensions ◽  
Maximum Density ◽  
Integer Lattice ◽  
Free Set ◽  
Integer Lattice Point ◽  
Positive Integers ◽  
Free Sets

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements $\mathbf{a}_1, \ldots, \mathbf{a}_k$ in $S$ satisfying $\mathbf{a}_1+\cdots+\mathbf{a}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions. 

On the Density of Universal Sum-Free Sets

1999 ◽  
Vol 8 (3) ◽  
pp. 277-280 ◽  
Author(s):  
TOMASZ SCHOEN
Keyword(s):  
Free Set ◽  
Free Sets

A set A is called universal sum-free if, for every finite 0–1 sequence χ = (e1, …, en), either(i) there exist i, j, where 1[les ]j<i[les ]n, such that ei = ej = 1 and i − j∈A, or(ii) there exists t∈N such that, for 1[les ]i[les ]n, we have t + i∈A if and only if ei = 1.It is proved that the density of each universal sum-free set is zero, which settles a problem of Cameron.

scholarly journals Maximal sum-free sets in finite abelian groups, V

1975 ◽  
Vol 13 (3) ◽  
pp. 337-342 ◽  
Author(s):  
H.P. Yap
Keyword(s):  
Prime Divisor ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Free Set ◽  
Free Sets

Let λ(G) be the cardinality of a maximal sum-free set in a group G. Diananda and Yap conjectured that if G is abelian and if every prime divisor of |G| is congruent to 1 modulo 3, then λ(G) = |G|(n−1)/3n where n is the exponent of G. This conjecture has been proved to be true for elementary abelian p−groups by Rhemtulla and Street ana for groups by Yap. We now prove this conjecture for groups G = Zpq ⊕ Zp where p and q are distinct primes.

COMPUTING MANY MAXIMAL INDEPENDENT SETS FOR HYPERGRAPHS IN PARALLEL

2007 ◽  
Vol 17 (02) ◽  
pp. 141-152 ◽  
Author(s):  
LEONID KHACHIYAN ◽  
ENDRE BOROS ◽  
VLADIMIR GURVICH ◽  
KHALED ELBASSIONI
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Average Degree ◽  
Maximal Independent Sets

A hypergraph [Formula: see text] is called uniformly δ-sparse if for every nonempty subset X ⊆ V of vertices, the average degree of the sub-hypergraph of [Formula: see text] induced by X is at most δ. We show that there is a deterministic algorithm that, given a uniformly δ-sparse hypergraph [Formula: see text], and a positive integer k, outputs k or all minimal transversals for [Formula: see text] in O(δ log (1 + k) polylog (δ|V|))-time using |V|O( log δ)kO(δ) processors. Equivalently, the algorithm can be used to compute in parallel k or all maximal independent sets for [Formula: see text].

scholarly journals Maximal sum-free sets in abelian groups of order divisible by three: Corrigendum

1972 ◽  
Vol 7 (2) ◽  
pp. 317-318 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Arithmetic Progression ◽  
Abelian Groups ◽  
Factor Group ◽  
Image Position ◽  
Free Set ◽  
Free Sets

The last step of the proof in [2] was omitted. To complete the argument, we proceed in the following way. We had shown that H = H(S) = H(S+S) = H(S-S), that |S-S| = 2|S| - |H| and hence that in the factor group G* = G/H of order 3m, the maximal sum-free set S* = S/H and its set of differences S* - S* are aperiodic, withso thatBy (1) and Theorem 2.1 of [1], S* - S* is either quasiperiodic or in arithmetic progression.

scholarly journals A Gray code for cross-bifix-free sets

2015 ◽  
Vol 27 (2) ◽  
pp. 184-196 ◽  
Author(s):  
ANTONIO BERNINI ◽  
STEFANO BILOTTA ◽  
RENZO PINZANI ◽  
VINCENT VAJNOVSZKI
Keyword(s):  
Gray Code ◽  
Constant Factor ◽  
Free Set ◽  
Free Sets

A cross-bifix-free set of words is a set in which no prefix of any length of any word is the suffix of any other word in the set. A construction of cross-bifix-free sets has recently been proposed in Cheeet al.(2013) within a constant factor of optimality. We propose a Gray code for these cross-bifix-free sets and a CAT algorithm generating it. Our Gray code list is trace partitioned, that is, words with zero in the same positions are consecutive in the list.

scholarly journals UPPER BOUNDS FOR SUNFLOWER-FREE SETS

2017 ◽  
Vol 5 ◽  
Author(s):  
ERIC NASLUND ◽  
WILL SAWIN
Keyword(s):  
Arithmetic Progression ◽  
Matrix Multiplication ◽  
Upper Bounds ◽  
Polynomial Method ◽  
Combinatorial Properties ◽  
Image Position ◽  
Free Set ◽  
Free Sets ◽  
Exponentially Small

A collection of $k$ sets is said to form a $k$-sunflower, or $\unicode[STIX]{x1D6E5}$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets ${\mathcal{F}}$sunflower-free if it contains no $3$-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), 339–343); (‘Progression-free sets in $\mathbb{Z}_{4}^{n}$ are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdős–Szemerédi sunflower problem (Erdős and Szemerédi, ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family ${\mathcal{F}}$ of subsets of $\{1,2,\ldots ,n\}$ has size at most $$\begin{eqnarray}|{\mathcal{F}}|\leqslant 3n\mathop{\sum }_{k\leqslant n/3}\binom{n}{k}\leqslant \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.\end{eqnarray}$$ We say that a set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}=\{1,2,\ldots ,D\}^{n}$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in A$ there exists a coordinate $i$ where exactly two of $x_{i},y_{i},z_{i}$ are equal. Using a version of the polynomial method with characters $\unicode[STIX]{x1D712}:\mathbb{Z}/D\mathbb{Z}\rightarrow \mathbb{C}$ instead of polynomials, we show that any sunflower-free set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}$ has size $$\begin{eqnarray}|A|\leqslant c_{D}^{n}\end{eqnarray}$$ where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$. This can be seen as making further progress on a possible approach to proving the Erdős and Rado sunflower conjecture (‘Intersection theorems for systems of sets’,J. Lond. Math. Soc. (2) 35 (1960), 85–90), which by the work of Alon et al. (‘On sunflowers and matrix multiplication’, Comput. Complexity22 (2013), 219–243; Theorem 2.6) is equivalent to proving that $c_{D}\leqslant C$ for some constant $C$ independent of $D$.

scholarly journals Dominating and unbounded free sets

1999 ◽  
Vol 64 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Slawomir Solecki ◽  
Otmar Spinas
Keyword(s):  
Closed Set ◽  
Analytic Set ◽  
Free Set ◽  
Free Sets

AbstractWe prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.

Almost Odd Random Sum-Free Sets

1998 ◽  
Vol 7 (1) ◽  
pp. 27-32 ◽  
Author(s):  
NEIL J. CALKIN ◽  
P. J. CAMERON
Keyword(s):  
Random Sum ◽  
Positive Probability ◽  
Finite Sum ◽  
Free Set ◽  
Positive Integers ◽  
Free Sets

We show that if S1 is a strongly complete sum-free set of positive integers, and if S0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S0 and is contained in S0∪S1. As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.

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