Symmetric complete sum-free sets in cyclic groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0, 1/3], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of ?_p of size at least (1/3?c)·p, where c > 0 is a universal constant.     

A complete solution to the infinite Oberwolfach problem

Let F be a 2‐regular graph of order v. The Oberwolfach problem, OP(F), asks for a 2‐factorization of the complete graph on v vertices in which each 2‐factor is isomorphic to F. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group G. We will also consider the same problem in the more general context of graphs F that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when F is locally finite. Moreover, we characterize the infinite subgraphs L of F such that there exists a solution to OP(F) containing a solution to OP(L).     

Bounds on the number of maximal sum-free sets

We show that the number of maximal sum-free subsets of {1,2,…,n} is at most 2^3n/8+o(n). We also show that 2^0.406n+o(n) is an upper bound on the number of maximal product-free subsets of any group of order n.     

On the number of maximal sum-free sets

It is shown that the set contains at most maximal sum-free subsets, provided is large enough.

     

Difference sets and sum-free sets in groups of order 16

Transversals for sum-free sets in the nine nonabelian order 16 groups are given. It is shown that exactly eight of the order 16 groups have difference sets Dwithλ = 2and D = -D. It is proven that the only (υ, k, λ) difference sets with υ = 2^t and λ = 2, have parameters (16, 6, 2). It is shown that exactly four of the order 16 groups can be partitioned into three sum-free sets.     

A new critical pair theorem applied to sum-free sets in Abelian groups

We shall prove a new generalization of Vosper critical pair theorem to finite Abelian groups. We next apply this new tool to the theory of $(k,l)$-free sets in finite Abelian groups. In particular, in most cases, we describe the structure of maximal $(k,l)$-free sets and determine the maximal cardinality of such a set. This result allows us for instance to give precisions on an old result of Yap: we are able to describe completely the maximal sum-free sets with cardinality at least one third of that of the ambient group.     

A complete solution of the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. We give a complete solution of this problem.     

Cyclic automorphisms of a countable graph and random sum-free sets

Cyclic automorphisms of the countable universal ultrahomogeneous graph are investigated using methods of Baire category and measure theory. This leads to the study of random sumfree sets; it is shown that the probability that such a set consists entirely of odd numbers is strictly positive, and bounds are given.     

A refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors,we show that if p is a sufficiently large prime and A is a sum-freesubset of the group of order p, such that n: = |A| > 0.318p,then A is contained in a dilation of the interval [n, p –n](mod p).     

On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x + y = z. In this paper, we provide a structural characterisation of sum-free subsets of {1, 2,..., n} of density at least 2/5 ? c, where c is an absolute positive constant. As an application, we derive a stability version of Hu’s Theorem [Proc. Amer. Math. Soc. 80 (1980), 711–712] about the maximum size of a union of two sum-free sets in {1, 2,..., n}. We then use this result to show that the number of subsets of {1, 2,..., n} which can be partitioned into two sum-free sets is Θ(2^4n/5), confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].     

Algebraic obstructions and a complete solution of a rational retraction problem

For each compact smooth manifold containing at least two points we prove the existence of a compact nonsingular algebraic set and a smooth map such that, for every rational diffeomorphism and for every diffeomorphism where and are compact nonsingular algebraic sets, we may fix a neighborhood of in which does not contain any regular rational map. Furthermore is not homotopic to any regular rational map. Bearing in mind the case in which is a compact nonsingular algebraic set with totally algebraic homology, the previous result establishes a clear distinction between the property of a smooth map to represent an algebraic unoriented bordism class and the property of to be homotopic to a regular rational map. Furthermore we have: every compact Nash submanifold of containing at least two points has not any tubular neighborhood with rational retraction.

     

A complete asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with geometric constraints

We consider an optimal control problem for solutions of a boundary value problem on an interval for a second-order ordinary differential equation with a small parameter at the second derivative. The control is scalar and is subject to geometric constraints. Expansions of a solution to this problem up to any power of the small parameter are constructed and justified.     

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2^{(1/2+∈(k))n} independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory. Research supported in part by the United States-Israel Binational Science Foundation and by a Bergmann Memorial Grant.     

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator on a normed lattice such that is not disjointness preserving.

     

An asymptotic solution to the cycle decomposition problem for complete graphs

Let m₁,m₂,…,mt be a list of integers. It is shown that there exists an integer N such that for all n?N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m₁,m₂,…,mt if and only if n is odd, 3?mi?n for i=1,2,…,t, and . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.     

A SECRETARY PROBLEM ON FUZZY SETS

This paper deals with a secretary problem on fuzzy sets, which allows both the recall of applicants and the uncertainty of a current applicant receiving an offer of employment. A new decision criterion is given to select a satisfactory applicant. This result extends the works of M.C.K. Yang and M.H. Smith.     

A solution to a problem of Dubins and savage

A problem left open in Dubins and Savages’ “How to Gamble if You Must” is solved. Research partially supported by NSF Grant DMS 91-57461.