Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.

Constructing Large Set Systems with Given Intersection Sizes Modulo Composite Numbers

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].