Kříž, Igor

Thomas, Robin

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeIgor Kříž; Robin D. ThomasOrdinal types in Ramsey theory and well-partial-ordering theoryCommentationes Mathematicae Universitatis Carolinae, Vol. 29 (1988), No. 1, 195--196Persistent URL: http://dml.cz/dmlcz/106610Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czANNOUNCEMENTS OF NEW RESULTS (of authors having an address in Czechoslovakia) MAXIMUM RANK OF A POWER OF A MATRIX OF A GIVEN PATTERN Svatopluk Poljak (Dept. of Applied Mathematics, Charles University, Malostran-ske" nam. 25, 11800 Praha 1, Czechoslovakia), received 17.10. 1987. Let G be a digraph with possible loops and without multiple edges. A t-walk is a sequence w=(v ,e,,v,,...,v. ,,e.,v.) of (not necessarily dist-inct) vertices and edges of G such that e;=V|_iV. is a directed edge for each i. We say that two t-walks w and w' are vertex (edge) independent if v. + v^ for i=0,...,t ( e ^ e ^ for i=l,...,t). A path is a walk with v.4-N . for i4*j, an(- a cycle is a walk with distinct vertices but v =v.. We denote by |P| and |C| the number of vertices of a path or a cycle. Theoren. For every digraph G and a positive integer p there are mutual-ly vertex disjoint cycles C,,...,C, and paths P,,...,P. such that the maximum number of vertex independent p-walks equals 2-_=ilc_l+ ^E i==:1C IP^ 1 —p). The above theorem may be interpreted as follows. Let m be the maximum number of people who could simultaneously walk in digraph G for p time units traversing one edge per a unit so that two or more people never meet in a vertex. Then the optimal schedule can always be organized as follows. The pe-ople are divided into several subgroups and each subgroup either walks round a cycle or along a path. Let A=(a..) be a real matrix of size n by n. The pattern of G is the di-graph on vertices •£l,2,...,nl and with an edge ij if a. -4*0. For a digraph G let A(G) be the class of matrices of pattern G. Corollary 1. Maximum possible rank of the p-th power Ap of a matrix A of a given pattern G equals maximum number of vertex independent p-walks in G. For a symmetric digraph G, we denote by <f(G) the class of symmetric matrices of pattern G. Corollary 2. maxAM<* r \ rAp=max._«/rNrAp for every symmetric digraph and a positive integer p. w t^ b ; K € * w The following Corollary 3 answers a question by J. Holenda who proved the case p=2. Corollary 3. max rAp=max r(A,,...,An) where A,A-,...,A„ are matrices of pattern G. p l p I thank J. Kratochvil for a valuable discussion about the problem. ORDINAL TYPES IN RAMSEY THEORY AND WELL-PARTIAL-ORDERING THEORY Igor KfXz (matematicko-fyzik^lni fakulta UK, Malostranske" nam. 25, 11800 Praha 1, Czechoslovakia), Robin Thomas (Matematicko-fyzikaTnf fakulta UK, Sokolovska' 83, 18600 Praha 8, Czechoslovakia), received 21.1. 1988. There is a big gap between the infinite Ramsey theorem (+) to~-+(cú)k and i ts f in i te version R ( n ; l 1 , . . . , l k ) - ~ - * Ü 1 , . . 1 )п - 195 The finite Ramsey theorem is much finer. We fill in the gap by defining "Ram-sey numbers'R(n; <y.,..., y k ) " for arbitrary ordinals T\^"^Tk' ^nese 9ene~ ralized Ramsey numbers are again ordinals, and their estimates provide a quan-titative strengthening of the infinite Ramsey theorem. There are different possible ways of measuring how sparse homogeneous sets are. For example, we may use the arithmetical density (this approach was studied by V. Rodi). Also our Ramsey numbers may be regarded as such measu-res, in fact, as somehow canohical ones: those of "finite character". In this sense, we are able to define Ramsey numbers in a much broader context than that of the classical Ramsey theorem mentioned above. We use a general axio-matical approach. The axioms themselves imply some estimates and other facts. For example, there are "non-uniform" and "uniform" Ramsey theorems and there is a broad class of cases when they coincide. To obtain sharper results, however, we have to consider more concrete situations. Those include beside the classical Ramsey theorem also the Erdos -Szeker£s theorem on monotone sequences, the Canonical (selective) Ramsey the-orem and the well-partial-ordering (WPO) theory. In the last case, the Ramsey numbers generalize the types of WPO sets, a concept already studied in great detail. More concretely, in the classical case we obtain upper and lower bounds analogous to (or slightly weaker than) those known in the finite case. This includes also the stepping-up lemma. Upper bounds for the "Canonical Ramsey numbers" are also obtained. The point of considering the Erdos-Szeker^s the-orem separately is that we have exact values or sharp estimates of the Ramsey numbers by certain product formulas. We also give a survey of the WPO theory from our point of view. Some known theorems with new and simpler proofs are included. Our Ramsey numbers are also closely connected with independence results in finite combinatorics. It has been already observed in the case of the ty-pes of WPO sets. To give a different example, the existence of "R(n;o>,... ...,c»>)" implies the Paris-Harrington modification of Ramsey theorem. As one might expect from unprovability of this theorem in PA, we have lim R(n;cu,...,o>)= t*. rtI,-+-Gi> o Finally, let us remark that our theory is different from the ordinal Ram-2 7 sey theorems (like e.g. o — + ( C J ,n)). From the philosophical point of view, our results are directed in the more "intensive" way. Their "extensive" power, for instance, in the classical case, does not exceed (+). 196 

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Ordinal types in Ramsey theory and well-partial-ordering theory

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Ordinal types in Ramsey theory and well-partial-ordering theory

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Institute of Mathematics AS CR, v. v. i.

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