András Hajnal

Hungarian Academy of Sciences | HAS · MTA Rényi Institute of Mathematics

Paul Erdős has published more than one hundred research papers in set theory. It is my rough estimate that these contain more than one thousand theorems, many having an interest in their own right. Although most of his problems and results have a combinatorial flavour, and the subject now known as “combinatorial set theory” is one he helped to crea...

A set-system $X$ is a $(\lambda, \kappa,\mu)$-system iff $|X|=\lambda$, $|x|=\kappa$ for each $x\in X$, and $X$ is $\mu$-almost disjoint. We write $[\lambda, \kappa, \mu] -> \rho$ iff every $(\lambda, \kappa,\mu)$-system has a "conflict free coloring with $\rho$ colors", i.e. there is a coloring of the elements of $\cup X$ with$\rho$ colors such th...

Partition relations were introduced in 1952 by Paul Erdős and Richard Rado to generalize Ramsey’s Theorem, yielding a seemingly inexhaustible supply of interesting problems. Unlike other classical problems these are far from being completely solved; indeed, there are only a few new deep results. We showcase modern methods of combinatorial set theor...

Given a function f a subset of its domain is a rainbow subset for f if f is one to one on it. We start with an old Erd˝os Problem: Assume f is a coloring of the pairs of !1 with three colors such that every subset A of !1 of size !1 contains a pair of each color. Does there exist a rainbow triangle ? We investigate rainbow problems and results of t...

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits \( \mathcal{T}_0 \) (the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can...

Volume 175, The Impact of Fertility Limitation on Woman's Life-Career and Pesonality pages 115–124, July 1970

Since the end of the 1950s László Kalmár has been interested in the information technology. During a 20 years period he designed several variants of computers interpreting high-level programming languages on architectural levels.

We present several partial results, variants, and consistency results concerning the following (as yet unsolved) conjecture. If X is a graph on the ground set V with $$Chr{\left( X \right)} = {\aleph }_{1}$$ then X has an edge coloring F with $${\aleph }_{1}$$ colors such that if V is decomposed into $${\aleph }_{0}$$ parts then there is one in whi...

We address partition problems of Erdös and Hajnal by showing that for all , if and carries a -dense ideal. If is measurable we show that for where is a very large ordinal less than that is closed under all primitive recursive ordinal operations.

It is shown that for any cardinal $\kappa, \dbinom{(2^{<\kappa})^{++}} {(2^{<\kappa})^+} \rightarrow \dbinom{\kappa}{\kappa}_{< \mathrm{cf} \kappa}$, and if $\kappa$ is weakly compact $\dbinom{\kappa^+}{\kappa} \rightarrow \dbinom{\kappa}{\kappa}_{< \kappa}$.

The relations M(K, λ, μ) → B [resp. B(σ)] meaning that if A ⊂ [K]λ with \A\ = K is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ρ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ρ. Then...

In this note, we prove the following: Theorem. On a countably infinite lattice L, there are 2 ℵ0 isotone maps. Proof. Let us assume that L does not have a unit element. Then it has a cofinal chain C of type ω, and so, for every infinite subset X of C, define ϕ X (a) as the smallest element of X that is ≥ a. The ϕ X provide continuumly many isotone...

The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a...

An increasing sequence of realsx=〈x i :i<ω〉 is simple if all “gaps”x i +1−x i are different. Two simple sequencesx andy are distance similar ifx i +1−x i <x j +1−x j if and only ify i +1−y i <y j +1−y j for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we pro...

It is consistent that there is an order type θ for which ψ(rightwards arrow eith stroke) [θ]2N1 holds for every type ψ.

Let the K p -independence number α p ( G ) of a graph G be the maximum order of an induced subgraph in G that contains no K p . (So K 2 -independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L 1 ,…, L r , we define the corresponding Turán-Ramsey function RT p ( n, L 1 ,…, L r , m ) to be the...

The set S⊂V() is a cut-set of the vertex v of a graph if v is not adjacent to any vertex in S and, for every maximal clique C of ,({ν}∪S)∩C≠Θ.S is a cut-set of the element v of a partial order if S is a cut-set of ν in the comparability graph of . Given upper bounds for the clique sizes and cut-set sizes of , we will determine the largest size of a...

We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3 and from probabilistic polynomial size circuits...

This paper is a continuation of [10], where P. Erds, A. Hajnal, V. T. Ss, and E. Szemerdi investigated the following problem:Assume that a so called forbidden graphL and a functionf(n)=o(n) are fixed. What is the maximum number of edges a graphG n onn vertices can have without containingL as a subgraph, and also without having more thanf(n) indepe...

We find here some extensions of the Erdős-Rado Theorem that answer some longstanding problems. Ordinary partition relations for cardinal numbers are fairly well understood (see [5]), but for ordinal numbers much has been open, and much remains open. For example, any proof of the simplest version of the Erdős-Rado Theorem seems to yield $$ For any r...

This paper is primarily a survey of recent results of the author and others on transfinite generalizations of Nešetřil-Rödl type Ramsey theory. In the second half of the paper large universal graphs are studied in detail.

We prove (in ZFC) the following theorem. Assume κ is an infinite cardinal, X is a Hausdorff space such that every subspace Y of X is the union of κ compact subsets of Y. Then X has cardinality at most κ.

We show that a κ-fold cover of a linearly ordered set by intervals is a union of κ disjoint covers.

A set T is said to cover a set system if T meets all members of . We raise the following general problem. Find relations among the natural numbers p, r, s, t, that imply the truth of the following statement: If . is an r-uniform set system such that each of its subsystems on at most p elements can be covered with an s-element set, then . can be cov...

It is shown that any graph onn vertices containing no clique and no independent set ont + 1 vertices has at least 2n \mathord/ \vphantom n (2t20 log(2t) ) (2t20 log(2t) ) 2^{{n \mathord{\left/ {\vphantom {n {(2t^{20 \log (2t)} )}}} \right. \kern-\nulldelimiterspace} {(2t^{20 \log (2t)} )}}} distinct induced subgraphs.

We prove that for any cardinalτ and for any finite graphH there is a graphG such that for any coloring of the pairs of vertices ofG withτ colors there is always a copy ofH which is an induced subgraph ofG so that both the edges of the copy and the edges of the complement of the copy are monochromatic.

We carry out the task given by the title, introduce a combinatorial principle, and use it to prove for all spaces for all spaces X where Y is any nondiscrete countable space, and related results.

Let G be a graph on n vertices, i(G) the number of pairwise non-isomorphic induced subgraphs of G and k≥ 1. We prove that if i(G) = o(nk+1) then by omitting o(n) vertices the graph can be made (l, m)-almost canonical with l + m ≤ k + 1.

In this paper we will consider Ramsey-type problems for finite graphs, r-partitions and hypergraphs. All these problems ask for the existence of large homogeneous (monochromatic) configurations of a certain kind under the condition that the size of the underlying set is large. As it is quite common in Ramsey theory, most of our results are not shar...

Let G be a graph on n vertices, i(G) the number of pairwise non-isomorphic induced subgraphs of G and k⩾1. We prove that if i(G)=o(nk+1) then by omitting o(n) vertices the graph can be made (l,m)-almost canonical with l+m⩽k+1.

If X is a graph, /c a cardinal, then there is a graph Y such that if the vertex set of Y is /c-colored, then there exists a monocolored induced copy of X; moreover, if X does not contain a complete graph on a 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 s...

We prove &thgr;(n log n) bounds for the deterministic 2-way communication complexity of the graph properties CONNECTIVITY, s-t-CONNECTIVITY and BIPARTITENESS (for arbitrary partitions of the variables into two sets of equal size). The proofs are based on combinatorial results of Dowling-Wilson and Lovász-Saks about partition matrices using the Möbi...

(of type co) on K there exists a free set of size A. These relations have been widely investigated so far, we do not give a complete history here. The interested reader is urged to meet [6] or [7]. We mention, however, the following results relevant to the problems we are going to consider. It was proved already in [5] ( as a consequence of the Erd...

The regressive partition relation, which turns out to be important in incompleteness phenomena, is completely characterized in the transfinite case. This work is related to Schmer [ S ], whose characterizations we complete.

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)⩽r2rlog2log2m (and this value is the best possible in a certain sense). We co...

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG 0,G 1 with χ(G 0)=χ(G 1)=ϰ+ and χ(G 0G 1)=ϰ. We also prove a result from the other direction. If χ(G 0)≧≧ℵ0 and χ(G 1)=kG 0G 1)=k.

A space X is said to have property P1(P2) if X is T2 and for any two disjoint open sets G and H in X we have |G| ⩽ ω and |H| ⩽ ω (|G| ⩽ ωor |H| ⩽ ω). We show that (i) if XϵP2 then |X| ⩽ 2ω; (ii) there is an XϵP1 with |X| = 2ω; (iii) there is a 0-dimensional (hence T3) space XϵP2 with |X| = 2ω; (iv) there is a compact XϵP2 with |X| = ω1; (v) it is c...

We wrote many papers on these subjects, some in collaboration with Galvin, Rado, Shelah and Szemerědi, and posed many problems some of which turned out to be undecidable. In this survey we state some old and new solved and unsolved problems.RésuméNous avons écrit beaucoup d'articles, certains en collaboration avec Galvin, Rado, Shelah et Szemerědi,...

We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.

The paper deals with game-theoretic versions of the partition relations $\alpha \rightarrow (\beta)^{

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 App...

The paper deals with common generalizations of classical results of Ramsey and Turán. The following is one of the main results. Assumek≧2, ε>0,G n is a sequence of graphs ofn-vertices and at least 1/2((3k−5) / (3k−2)+ε)n 2 edges, and the size of the largest independent set inG n iso(n). LetH be any graph of arboricity at mostk. Then there exists...

This chapter discusses problems related to bipartite large chromatic graphs. It is assumed that the chromatic number χ(Ң) of a graph Ң is greater than κ, a finite or infinite cardinal. This problem is investigated in the chapter, in case some other restrictions are imposed on Ң as well. The results show that χ(Ң) can be arbitrarily large while the...

A space X is said to satisfy condition (C) if for every Y⊂X with |Y|=ω1, any family of open subsets of Y with ||=ω1 has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace [X] is CCC. We show that under MAω1 condition (C) is also necessary for [X] to be CCC, but under CH it is not.

Papers Presented at a Colloquium organized by the János Society, June 25 July 1, 1973

We show that for every cardinal $\kappa > \omega$ and an arbitrary topological space $X$ if we have $w(Y)

This chapter provides solutions to two problems: (1) producing under continuum hypothesis (CH) or by adding lots of Cohen reals a regular space X with R (X)=ѡ < nw (X) = 2ѡ and (2) using Martin's axiom (MA) (ѡ1)+◊ѡ2 (E), where E= { a∈ѡ2: cf (a)=ѡ}, a Hausdorff space X such that nw(X) > ѡ but nw (Y) = ѡ whenever YCX, | Y|=ѡ1. The chapter shows that...

There are many facts known about the size of subsets of certain kinds in free lattices and free products of lattices. Examples: every chain in a free lattice is at most countable; every “large” subset contains an independent set; if the free product of a set of lattices contains a “long” chain, so does the free product of a finite subset of this se...

This chapter examines the tightness of the topological product of two spaces of tightness ѡ. The chapter proves that if the axiom of constructibility holds, then for each cardinal к there are two Frèchet–Urysson spaces such that the product space has tightness к. Assuming that the continuum is relatively small, it can prove that there are many card...

Several applications of the Cech-Pospisil theorem are given; one of them states (under CH) that every uncountable compact space has a Lindelof subspace of cardinality ω1.

Applying the continuum hypothesis, we construct a hereditarily separable and hereditarily normal topological group which is not Lindelöf.

This chapter discusses some aspects of combinatorial set theory related to recent results. The chapter presents results in their simplest forms so that their proofs reflect the main ideas and difficulties to be found in the proofs of analogous but more comprehensive results. The chapter also discusses chromatic numbers of a certain kind of graphs.

In this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscript should not exceed two doublespaced type-written pages.

The aim of this note is to show, without using any special set-theoretic assumptions, that the product of two (weakly) Lindelöf spaces is not necessarily weakly Lindelöf.

In this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscript should not exceed two doublespaced typewritten pages.

In this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscript should not exceed two doublespaced typewritten pages.

A family of k-subsets of an n-set such that no more than r have pairwise fewer than s elements in common is maximum (for sufficiently large n) only if consists of all the k-sets containing at least one of r fixed disjoint s-subsets.

In this column Periodica Mathematica Hungarica intends to publish current research problems whose proposers believe them to be within reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscript should not exceed two doublespaced typewritten pages...

In this paper we shall prove some results about tournaments which we believe to be interesting both from an algebraic and a set theoretic point of view. The definition of a simple tournament, the subject of our title, was motivated by questions in algebra, but the results and the proofs we give are essentially set theoretical. We assume that the re...

holds. Here x → y expresses the fact that {x, y} → and we sometimes write this in the alternative form y ← x. Extending the notation to subsets of T we write A → B or B ← A if a → b holds for all pairs a, b with a A and b B. is a subtournament of , and is an extension of , if T′ T and →′ is the restriction of → to T′; we will usually write ′, → ins...

The aim of this paper is to evaluate the spread of a product with the help of the spreads of its factors and their number. The main result says that if finitely many T2 spaces are such that none of them contains a discrete sunspace of power >α, then their product does not contain a discrete subspace of power >2α.

This chapter provides an introduction to set-theoretic assumptions, such as Martin's axiom and ◇, unearthed by set theorists and illustrates how these can be used in proving topological theorems or constructing counter examples. The notation used in the chapter follows JUHÁSZ; one exception is the tightness that is denoted by t(X). The chapter also...

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 commu...