On topological properties of ultraproducts of finite sets

In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. Here we present the solutions of two such problems. More concretely we show 1. that there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and 2. if A is an infinite ultraproduct of finite sets, then every ultratopology on A contains a dense subset D such that |D| < |A|. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On intersecting families of finite sets

Let X = [1, n] be a finite set of cardinality n and let $\text{F}$ be a family of k-subsets of X. Suppose that any two members of $\text{F}$ intersect in at least t elements and for some given positive constant c, every element of X is contained in less than c |$\text{F}$| members of $\text{F}$. How large |$\text{F}$| can be and which are the extremal families were problems posed by Erdös, Rothschild, and Szemerédi. In this paper we answer some of these questions for n > n₀(k, c). One of the results is the following: let $\text{t = 1,}\text{3}\text{7}\text{< c <}\text{1}\text{2}$. Then whenever $\text{F}$ is an extremal family we can find a 7-3 Steiner system $\text{B}$ such that $\text{F}$ consists exactly of those k-subsets of X which contain some member of $\text{B}$.     

On maximal intersecting families of finite sets

Let r be a positive integer. A finite family $\text{H}$ of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ? $\text{H}$, |A| ? r there exists a member E ? $\text{H}$ such that $\text{A ∩ E = ?}$. For instance, a finite projective plane of order r ? 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove $\text{N(r) ?}\text{3}\text{4}\text{r}{}^2$ whenever a projective plane of order $\text{r}\text{2}$ exists. This disproves the known conjecture N(r) ? r² ? r + 1.     

On separation properties of finite dimensional compact convex sets

In a real finite-dimensional vector space, we study families of sets such that every compact convex set in the space is the intersection of all members of the family that contains it.

     

Local topological properties of complex algebraic sets

Siberian Mathematical Journal -     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

On ramsey families of sets

The main result of this paper is a lemma which can be used to prove the existence of highchromatic subhypergraphs of large girth in various hypergraphs. In the last part of the paper we use amalgamation techniques to prove the existence for everyl, k 3 of a setA of integers such that the hypergraph having as edges all the arithmetic progressions of lengthk inA has both chromatic number and girth greater thanl.     

On families of intersecting sets

While algorithms exist which produce optimal binary trees, there are no direct formulas for the cost of such trees. In this note, we give a formula for the cost of the optimal binary tree built on m nodes with weights 1, 2, 3,…, m. The simplicity of this proof suggests that one can try to compute the cost of optimal trees for other special classes of weights.     

Approximating properties of sets in some topological spaces

We investigate the approximating properties of sets in spaces normed over semifields. We establish criteria for sets to possess the minimization property in several classes of spaces. A set possesses this property if, for every element of the space, there is a sequence of elements of the set which give arbitrarily close approximations to the best approximation. 5 titles.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 523–530, November, 1967.     

On the topological structure of univoque sets

Erd?s, Horváth and Joó discovered some years ago that for some real numbers 1<q<2 there exists only one sequence ci of zeroes and ones such that ∑ciq−i=1. Subsequently, the set U of these numbers was characterized algebraically in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions 1=∑q−ni and related problems, Bull. Soc. Math. France 118 (1990) 377-390] and [V. Komornik, P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar. 44 (2) (2002) 195-216]. We establish an analogous characterization of the closure of U. This allows us to clarify the topological structure of these sets: is a countable dense set of , so the latter set is perfect. Moreover, since U is known to have zero Lebesgue measure, is a Cantor set.     

On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets.

     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.     

On additive properties of product sets in an arbitrary finite field

It is proved that for any two subsets A and B of an arbitrary finite field $ \mathbb{F}_q $ \mathbb{F}_q such that |A||B| > q, the identity 10AB = $ \mathbb{F}_q $ \mathbb{F}_q holds. Under the assumption |A||B| ⩾2q, this improves to 8AB = $ \mathbb{F}_q $ \mathbb{F}_q .     

Some topological properties of support points of convex sets

It is shown, as a corollary to Trojanski’s renorming theorem, that the set of support points (of certain closed convex sets) is connected. Some connectedness properties are established for support functionals, and it is also shown that the support points form an F_σ.