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\-v coLLoQUlA *MATHEMATICA* *SOCIETATIS* JANOS *BOLYAI* 5. HILBERT SPACE OPERATORS. TIHANY (HUNGARY), 1970 Symbols of operators and quantization

"... Let L z ( M) be the Hilbert space of functions square-integrable on a measure space M.It is convenient to define linear operators on L2(M)bymeansof func-'ons of two variables. The best known way is by means of kernel: (Af)(r) : Jxc*,9)f t9)dg. Incase of suchacorrespondencebetweenoperators A and ..."

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Let L z ( M) be the Hilbert space of functions square-integrable on a measure space M.It is convenient to define linear operators on L2(M)bymeansof func-'ons of two variables. The best known way is by means of kernel: (Af)(r) : Jxc*,9)f t9)dg. Incase of suchacorrespondencebetweenoperators A and functions 9(x,9), we shall call the function g(r,U) the symbol of the corresponding operator A. The kernel K {x, g) is the particular case of symbol. In this paper we give a survey on different sorts of symbols, considering mainly the case when M is the real n-dimensional euclidean space with the usual Lebesque measure. One of the sources of appearance of symbols is the expression of operators in algebras of operators through generators 0 u and Q-n, where 0u and Q1 are the "operators of impulse and coordinate " well known from quantum mechanics:

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COLLOQUIA *MATHEMATICA* *SOCIETATIS* JANOS *BOLYAI* 37. FINITE AND INFINITE SETS, EGER (HUNGARY), 1981. SIZE RAMSEY NUMBERS INVOLVING MATCHINGS

"... Let F, G and H be finite, simple and undirected graphs. The edges and number of edges of a graph F will be denoted by E(F) and I E(F) I respectively. A graph F- (G, H) if every 2-coloring (say red and blue) of E(F) produces either a "red " G or a "blue " H. The size Ramsey number ..."

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Let F, G and H be finite, simple and undirected graphs. The edges and number of edges of a graph F will be denoted by E(F) and I E(F) I respectively. A graph F- (G, H) if every 2-coloring (say red and blue) of E(F) produces either a "red " G or a "blue " H. The size Ramsey number r(G, H) = min f l E(F) 1: F-> (G, H)}. For t> 1, the graph consisting of t independent edges will be denoted by tK2. In this paper, bounds and in some cases exact values will be calculated for r(tK2, G) for various classical graphs G, for example, when G is either a small order graph, a path, a cycle, a complete graph or a complete bipartite graph. Asymptotic results are obtained for some graphs in which exact values could not be calculated. 1

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COLLOQUIA *MATHEMATICA* *SOCIETATIS* JANOS *BOLYAI* 4. COMBINATORIAL THEORY AND ITS APPLICATIONS, BALATONFURED (HUNGARY), 1969. On a lemma of Hajnd-Folkman bY

"... proved the following Lemma: Let ISI = b-f, A; c 9, IA;1 = n be subsets of 9 so that to every element x of 9 there is an A; not containing x. We define now a graph as follows: XE 9, y e 9 are joined if for some A; they are both contained in A;. The Lemma asserts that this graph contains a complete gr ..."

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proved the following Lemma: Let ISI = b-f, A; c 9, IA;1 = n be subsets of 9 so that to every element x of 9 there is an A; not containing x. We define now a graph as follows: XE 9, y e 9 are joined if for some A; they are both contained in A;. The Lemma asserts that this graph contains a complete graph of n+ ( vertices. We are going to generalise and extend this Lemma in various directions and establish some connections with RAMSEY ’ s theorem. First we have to introduce some notations. The basic elements of an r-graph are its vertices and the r-tuples formed from some of its vertices. Kr ( n) is the complete P-graph of n vertices and all its (:) r-tuples. For r-2 we obtain the ordinary graphs. Let 9 be a set. A family of subsets A; c 9 defines an r-graph f+c’) ( 9; A,)...) as follows: The vertices are the elements of 9, an r-tuple belongs to our graph if and only if it is a subset of one of our A’s. Such r-graphs were, as far as I know, first studied in [l] in a context that differs from this. We say that the family can be- 311-represented by i vertices if there are i elements Y,,.. 7 xi of 9, sothat

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COLLOQUIA *MATHEMATICA* *SOCIETATIS* JP;NOS *BOLYAI* 10. INFINITE AND FINITE SETS, KES~THELY (HUNGARY), 1973. EUCLIDEAN RAMSEY THEOREMS, II

"... We fist recall a few definitions and results from [ 21. Let K be a set of points in Euclidean n-space, En, and let the points of En be r-colored (that is, divided into Y classes, or colors). Then if all the points of K are the same color (i.e., in the same class), K is said to be monochroma ..."

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We fist recall a few definitions and results from [ 21. Let K be a set of points in Euclidean n-space, En, and let the points of En be r-colored (that is, divided into Y classes, or colors). Then if all the points of K are the same color (i.e., in the same class), K is said to be monochroma

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COLLOQUIA *MATHEMATICA* *SOCIETATIS* JÁNOS *BOLYAI* 34.TOPICS IN CLASSICAL NUMBER THEORY BUDAPEST (HUNGARY), 1981. SOME RESULTS IN COMBINATORIAL NUMBER THEORY

"... We should like to state here some new results in combinatorial number theory. 1. In [31 p.50 the following question was asked: "Let A={a1<a2<...} and B={b1<b2<...I be sequences of integers satisfying A(x)> Ex l/2, B(x)> Ex l/2 for some e>O. Is it true that (1) a í-- a.7 = b ..."

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We should like to state here some new results in combinatorial number theory. 1. In [31 p.50 the following question was asked: "Let A={a1<a2<...} and B={b1<b2<...I be sequences of integers satisfying A(x)> Ex l/2, B(x)> Ex l/2 for some e>O. Is it true that (1) a í-- a.7 = bk- b t has infinitely many solutions? " (A(x) and B(x) are the number of elements of A and B up to x, resp.) R. Freud observed that the answer was negative: we write the numbers in binary scale, and select for A those ones which contain only even powers of two, and forB those which contain only odd powers of two. Then (1) is possible only in the trivial case, and m = lim inf min{A(x), B(x)} = 1 x--Later P. Erdős and R. Freud investigated general properties of sequences A and B for which (1) has only trivial solutions. We state here some of these results: 1.1. m> 2-1/4- e is attainable. 1.2. The largest possible value of s = lim sup A(x)B(x)/x is 2. More precisely, if x--lim sup x(n) _ then A(n)B(n) ? 2n-x(n) is attainable n for infinitely many n by suitable A and s, but A(n)B(n)-2n--- for any A and s. 1.3. s = lim inf A(x)B(x)/x is at most 14/9, x--more precisely (5/2)s + 2s 5 7. Also s + ( 3/2)s < 4, which shows that s=2 implies s<_1. It is not yet known whether s>l is possible at all.1.4. If m>o, then neither A(x)/J; nor B(x)/fx can tend to a limit. Several further theorems are proved on the behaviour of A(x)B(x)/x, A(x)/,lx and B(x)/fx. The results with detailed proofs will appear in Ell, and another forthcoming paper will deal with related problems 2. Now we consider permutations of integers. In the finite case let al,a2,...,an be a permutation of the integers 1,2,...,n, and in the infinite case let al,a2,...,ai,... be a permutation of all positive integers P.Erdős, R.Freud and N.Hegyvári investigated several estimations concerning the values of [a i ~ a i+l~ and (a i la i+l). 2.1. In the finite case

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COLLOQUIA *MATHEMATICA* *SOCIETATIS* J/iNOS *BOLYAI* IO. INFINITE AND FINITE SETS, KESZTHELY (HUNGARY), I 973. A NON-NORMAL BOX PRODUCT

"... We use the convention that a cardinal is the smallest ordinal of that cardinality, and an ordinal is the set of ordinals less than it is. The topology on an ordinal is the order topology, If %AEw * is a collection of topological spaces, then the box product of IXnInoO is 17 Xn with the topology indu ..."

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We use the convention that a cardinal is the smallest ordinal of that cardinality, and an ordinal is the set of ordinals less than it is. The topology on an ordinal is the order topology, If %AEw * is a collection of topological spaces, then the box product of IXnInoO is 17 Xn with the topology induced by using FlEWf) ( 17 n’wo

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