Gyula O. H. Katona

A theorem of finite sets

HAJNAL proved this statement in the case of l = 3 (unpublished). In this paper I prove for all cases that this is, indeed, the minimum, and find the (more complicated) minimum also for arbitrary n. The theorem is probably useful in proofs by induction over the maximal number of elements of the subsets in a system, as was SPERNER’S, lemma in his paper [1].

Contributions to the geometry of hamming spaces

Let 1 ⩽ N ⩽ 2n and let U, B denote families of subsets of {1,…n}. The following results are proved: • Theorem 2.3 (U, B) is a d-pair, 1 ⩽ d ⩽ n, if |AΔB| ⩽ d for all A ϵ Y, B ϵ B. Then max{|B|: (U, B) is d-pair and |B = N} is assumed if Y is a “quasi-sphere”. • Theorem 3.1. min|AþB| is assumed for a (pseudo)-sphere characterized by the property that ‖{ A : A ϵ U, x ϵ A|-| A : A ϵ U,y ϵ A}‖ ⩽ 1 for all x, y ϵ 1,2,…,n. Denote by K1 = K1(U)(i = 0,1,…,n) the numberof i-element members of an order idealU. • Theorem 4.2. minn n-NΣ1K1W1 is assumed 1. (a) in case W0 ⩽ W1 ⩽ ⋯ ⩽ Wn if U is a quasi-spere. 2. (b) in case W0⩾W1⩾⋯⩾Wnif Uis a quasi-cylinder. • Theorem 4.5.minnn-NΣ1K1W1is assumed 1. (a)in case W0⩽W1 ⩽ ⋯ WM + 1 ⩾ ⋯ ⩾ Wn by a union of a quasi-cylinder and a quasi-sphere, 2. (b) in case W0 ⩾ W1 ⩾ ⋯ ⩾WM⩽ WM+1 ⩽ ⋯ ⩽ Wn by an intersection of a quasi-cylinder and a quasi-sphere.

Extremal Problems for Hypergraphs

By a hypergraph we mean a pair (V,A), where V is a finite set, and A = {A1,…,Am} is a family of its different subsets. |V| means the number 1 m of elements of V; this is usually denoted simply by n. Similarly, |A| =m. The elements of V are called vertices, the elements of A are the edges.

Combinatorial Search Problems

Publisher Summary This chapter discusses combinatorial search problems. There are many practical problems of this type. Wasserman-type blood test of a large population is such a problem. X is the set of some men. The test can be divided into two parts: (1) a sample of blood is drawn from every man and (2) the blood sample is subjected to a laboratory analysis that reveals the presence or absence of syphilitic antigen. The presence of syphilitic antigen is a good indication of infection; for the second step, instead of carrying out the test individually some samples can be poured together. Diagnosis of a sick TV set is another problem. X is the set of parts of the TV set. First one sees that there is a good picture. The trouble should be in the sound-channel, which is a subset of the set of parts of the TV set. Similarly, by different tests, it can be determined whether certain subsets contain the ill part or not. Chemical analysis is another problem. The third problem explains how to identify an unknown chemical element. X is the set of chemical elements. Pour some chemical to the unknown one; if its color turns red, then it belongs to a subset of the set of chemical elements; in the contrary case, it does not. After carrying out some such tests, the unknown element can be identified.

On separating systems of a finite set

Abstract Let H be a finite set, and A 1 , A 2 , …, A m subsets of H . We call a system A separating system, if for any two distinct elements x and y of H there exists an A i (1≤ i ≤ m ) such that either. x ∈ A i and y ∉ A i or x ∉ A i and y ∈ A i This paper deals with the problem of finding the minimum of m , if additionally ⋎ A i ⋎≤ k (1≤ i ≤ m ), where 1≤ k ≤ n , and ⋎ A i ⋎ is the cardinal number of A i . We reduce this combinatorial problem to an analytical one, and give a lower and an upper estimation: log ⁡ n log ⁡ en / k n k ≤ min ⁡ m ≤ { log ⁡ 2 n log ⁡ n / k } n k

Semantics in databases

The term “Semantics” is one of the overloaded in computer science and used in various meaning. This variety can also be observed in database literature. In computer linguistics or web research, semantics is a component of the language which associates words or components of a grammar with their meaning (linguistic content). In modeling and specification, semantics assigns set-theoretic denotations to formulas in order to characterize truth. At the same time, semantics is used as the basis for certain methods of proof (truth and proof semantics in semiotics). In programming language technology, semantics is often used in the sense of operational semantics, i.e. consists in an interpretation of commands of a programming language by machine operations. This widespread usage of the term “semantics” has led to very different goals, methods, and applications. Semantics includes at the same time the interpretation of utterances, temporal, contextual, subjective and other aspects. Semantics is either considered operationally on the basis of applications or implementations, or logically associating a database state or a collection of database states to a truth value or pragmatically by relating utterances to the understanding of the user. These three understandings may be mapped to each other.

Largest Families Without an r-Fork

Let [n] = { 1,2,...,n} be a finite set,

Largest family without A ∪ B ⊆ C ∩ D

{\cal F}

No four subsets forming an N

a family of its subsets, 2 ≤ r a fixed integer. Suppose that

A generalization of some generalizations of Sperner's theorem

{\cal F}