Andras Hajnal

On the maximal number of independent circuits in a graph

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .

Partition relations for cardinal numbers

1. INTRODUCTION In this paper our main object is the study of relations between cardinal numbers which are written in the form a-(b o , b,,. . .) r or a-(b)C ~° or (b) (b o b,) Such relations were introduced in [3] and [1]. They are called I-relations, Ii-relations and III-relations respectively, and they will be defined in 3. 1, 3. 2 and 3. 3. In sections 18 and 19 we shall introduce certain generalizations of these relations. Our whole theory can be considered as having arisen out of the classical theorem of RAMSEY [18]. The most natural and direct generalization of the question settled by Ramseys theorem is the problem of deciding whether for a given positive integer r and given cardinals a, b o , b,,. .. the I-relation a-(b o , b,,. . .)r is true or false. Although the I-relation was only introduced in 1952 several papers had already appeared on such relations between 1933 and 1952. As far as we know the first of these papers is due to W. SIERPINSKI [19] who proved that 2lá-(N,, N,) 2. Two of the present authors have published several notes on such relations. All the results found by us before the present paper are contained in [1] where there will also be found references to previous papers by other writers except to those of G. KUREPA of which we had no knowledge at that time. Independently of P. ERDŐS and R. RADO and at about the same time KUREPA found several I-relations the most important of which, deduced under the assumption of the generalized continuum hypothesis, can be stated as follows : (~a+2-1~a+1, K+2 2 , 1~«+r~(«+1) r r, for r 1, a 0. Furthermore, KUREPA proved independently but somewhat later than SIERPINSKI that 2 +(a+1, a+1) 2. For these results see [20]. In this paper our first major aim is to discuss as completely as possible the relation I. Our most general results in this direction are stated in Theorems 1 and II, and the remaining open questions are stated in Problems 1 and 2. If we disregard cases when among the given cardinals there occur inaccessible numbers greater than t~ o , and if we assume the General Continuum Hypothesis, then our results are complete for r = 2, and they are also complete for r-3 provided we restrict ourselves to finitely many numbers b,. …

Combinatorial set theory : partition relations for cardinals

Fundamentals about Partition Relations. Trees and Positive Ordinary Partition Relations. Negative Ordinary Partition Relations and the Discussion of the Finite Case. The Canonization Lemmas. Large Cardinals. Discussion of the Ordinary Partition Relation with Superscript 2. Discussion of the Ordinary Partition Relation with Superscript < 3. Some Applications of Combinatorial Methods. A Brief Survey of the Square Bracket Relation.

A Problem in Graph Theory

A gruph consists of a finite set of vertices some pairs of which are adjacent, i.e., joined by an edge. No edge joins a vertex to itself and at most one edge joins any two vertices. The degree of a vertex is the number of vertices adjacent to it. The complete k-graph has k vertices and @ edges. We shall say that a graph G has property (n, k), where n and k are integers with 25 kSn, if G has n vertices and the addition of any new edge increases the number of complete k-graphs contained in G. For example, let Ar{n) denote a graph with n vertices and n(k-2)-(,I) edges which consist of a complete (k-2)-graph each vertex of which is also joined to each of the n-(k-2) remaining vertices. A&z) contains no complete k-graphs but it is easily seen that with the addition of any new edge a compIete k-graph is formed. Hence, &(n) has property b, k).

On a property of families of sets

1. Introduction. In this paper we are going to generalize a problem solved by MILLER in his paper [l] and prove several results concerning this new problem and some related questions. We mention here that some of our theorems (Theorems 8 and 10) have the interesting consequence that the topological product of Ks l-compact spaces (Lindelof spaces) is not necessarily

A cure for the telephone disease

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer. () The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal*! If f(n) is the minimum number of calls needed, then it is easy to verify that / ( l ) = 0 , / ( 2 ) = l , / ( 3 ) = 3 , / ( 4 ) = 4 . It is also easy to see tha t / («+l )^ / ( /z )+2 , for the (n+ l)-th lady first calls one of the others and someone calls her back after the remaining n ladies have communicated all the scandal to each other. It follows that f(n) 4). We will prove that

Coloring graphs with locally few colors

Abstract Let G be a graph, m > r ⩾1 integers. Suppose that it has a good coloring with m colors which uses at most r colors in the neighborhood of every vertex. We investigate these so-called local r -colorings. One of our results (Theorem 2.4) states: The chromatic number of G , Chr( G )⩽ r 2 r log 2 log 2 m (and this value is the best possible in a certain sense). We consider infinite graphs as well.

A tribute to Paul Erdős

Do you ever know the book? Yeah, this is a very interesting book to read. As we told previously, reading is not kind of obligation activity to do when we have to obligate. Reading should be a habit, a good habit. By reading, you can open the new world and get the power from the world. Everything can be gained through the book. Well in brief, book is very powerful. As what we offer you right here, this tribute to paul erdos is as one of reading book for you.

Embedding graphs into colored graphs

If X is a graph, K a cardinal, then there is a graph Y such that if the vertex set of Y is s-colored, then there exists a monocolored induced copy of X; moreover, if X does not contain a complete graph on ce vertices, neither does Y. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph X such that for every graph Y there is a two-coloring of the edges of Y such that there is no monoeolored induced copy of X. Similarly, a triangle-free X may exist sueh that every Y must contain an infinite complete graph, assuming that coloring Ys edges with countably many colors a monocolored copy of X always exists. 0. Introduction. In this paper we deal with the generalization of partition theory which investigates the existence of monocolored prescribed subgraphs of multicolored graphs satisfying certain conditions. As usual we will need partition symbols to make the formulation of the results and problems feasible. (0.1) Y ) (X)l X y > (X)2 mean that the following statements are true. If the vertices/edges of Y are ey-colored then there exists a monocolored copy of X c Y, respectively. (0.2) Y (X)1, Y (X)2 mean the existence of monocolored copies of X which are induced subgraphs of Y. Clearly, the Erdos-Rado generalization of Ramseys theorem yields an obvious existence theorem of type VX BY in (0.1), and the meaningful results concerning this symbol are of the form VX E BY E g for certain classes , g of graphs. The existence problem for the symbols (0.2) is nontrivial, though it is quite easy for the first symbol and here the problem has to be investigated under additional restrictions on X and Y. As to t symbol Y (X)2 the stat m nt VX BY Y (X)2 forey < ,X and Y) finite was proved by three diderent sets of authors [4, 10, 20] and it was extended for countable graphs X in [10] wher Vt W V0Xt < W HtY < 2w y (X)2 was proved. One of the main observations of this paper is that (contrary to the intuitive expectation of combinatorialists that this kind of Ramsey property always holds Received by the editors March 21, 1986 and, in revised form, April 15, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05; Secondary 03E35, 04A20, 05C65.

Some remarks concerning our paper „On the structure of set-mappings” —Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal

In accordance with the notations of [4] we say that a cardinal m possesses property P 3 if every two-valued measure μ(X) defined on all subsets of a set S of power m vanishes identically, provided p({x}) =0 for every x E S and p(X) is m-additive . It was well known that tt o fails to possess property P 3 and that every cardinal m < t, possesses property P 3 where t, denotes the first uncountable inaccessible cardinal . Recently A . TARSKI has proved, using a result of P. HANF, that a certain wide class of strongly inaccessible cardinals possesses property P 3 (called strongly incompact cardinals) . H . J . KEISLER gave a purely set-theoretical proof of this result . After having seen these papers we observed that the special case of this result that t, possesses property P 3 follows almost trivially from some of our theorems proved in (1] . We are going to give this simple proof in § 2 . Our method for the proof is of purely combinatorial character, and although it is certainly weaker than that of A. TARSKI and H . J. KEISLER, we think that it is of interest to formulate how far one can go with these methods at present . Let to , . . ., t,, . . . denote the increasing sequence of the strongly inaccessible cardinals (to = o) and let (O, denote the initial number of t4 . We can prove similarly as in the case of t, that i s possesses property P 3 , provided 0 < ~ 0 4. We only give the outline of this proof. Finally, we are going to formulate some problems . .