Ramsey Properties for Classes of Relational Systems

Discrete Mathematics, 1991

El-Zahar, M. and N-W. Sauer, Ramsey-type properties of relational structures, Discrete Mathematics 94 (1991) l-10. Let .=%'be a relational language and \!I be a set of 6P-structures. Vl is indivisible if for each A E '!I there is a relational structure R(A) E Vl such that for every partition of R(A) into two classes C and D, there is an embedding of A into C or into D. (If Folkman's Theorem (1970) hold in '?I). We will investigate this property of indivisibility in the case where '!I= age S for some countable relational structure S (age S is the set of all finite substructures of S up to isomorphism). In particular, if S is homogeneous, the divisibility or indivisibility of age S is related to the way in which the elements of age S amalgamate.

Journal of Combinatorial Theory, Series A, 1983

The purpose of this paper is to prove a strengthening of [4], which gives (practically) the full characterization of Ramsey classes of (ordered) set systems. We also provide a modified proof of the main result of [4]. Let us remark that [4] contains several technical mistakes but they can be corrected by changing details. Although the basic ideas of the proof here and of [4] are the same both differ at several places and we hope that the present proof is more direct and transparent. We leave out the motivation for this particular research (see e.g., [3, 5, and 61. Particular cases, which appeared e.g., in [ 1, 7-91 are simpler results. Presently, there is no other proof for the theorem presented here apart from kw. This paper is written in the finite set theory. To state the main result we need some preliminaries. A family d = (Si; i E I) of natural numbers is called a type. A set system of type d is a pair (X,-H), where X is a (totally) ordered set, M=(&;iEI) and McX, ]MI=6, for every ME&. The ordering of X is called the stundurd.ordering and usually denoted by <. We shall sometimes write explicitly ((X, <), M). Let A = (X, A) is said to be a subobject of B = (Y, JV), J"' = (4; i E I) if X is a monotone subset of Y (monotone with respect to the standard orderings) and 4 = (it4 E 4; A4 C X} for every i E I. Denote by Sot(d), the class of all set systems of type d together with all embeddings. The set of all subobjects of B, which are isomorphic to A, will be' denoted by (,"). The elements of the set (:) will be also called A-183

Cambridge University Press eBooks, 2018

We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Nešetřil-Rödl Theorem, the authors Ramsey lift of bowtie-free graphs as well as the Ramsey Theorem for Finite Models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of Structural Ramsey Theorem. We give here a more concise reformulation of the recent paper "All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)" and the main purpose of this paper is to show several applications. Particularly we prove the Ramsey property of ordered sets with equivalences on the power set, Ramsey theorem for Steiner systems, Ramsey theorem for resolvable designs and a partial Ramsey type results for H-factorizable graphs. All of these results are natural, easy to state, yet proofs involve most of the theory developed.

Discrete Mathematics, 1982

2003

We prove induced Ramsey theorems in which the induced monochromatic subgraph satisfies that some of its partial automorphisms extend to automorphisms of the colored graph.

Journal of Combinatorial Theory, Series B, 2002

For a graph F and natural numbers a 1 ; . . . ; a r ; let F ! ða 1 ; . . . ; a r Þ denote the property that for each coloring of the edges of F with r colors, there exists i such that some copy of the complete graph K ai is colored with the ith color. Furthermore, we write ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ if for every F for which F ! ða 1 ; . . . ; a r Þ we have also F ! ðb 1 ; . . . ; b s Þ: In this note, we show that a trivial sufficient condition for the relation ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ is necessary as well. # 2002 Elsevier Science (USA) # 2002 Elsevier Science (USA)

Transactions of the American Mathematical Society, 1987

It is shown that the class of partial Steiner (fc, Z)-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner (k, i)-system Q there exists a partial Steiner (fc, Z)-system)i such that for every partition of the edges of H into two classes one can find an induced monochromatic copy of Q. As an application we get that the class of all graphs without cycles of lengths 3 and 4 has the edge Ramsey property. This solves a longstanding problem in the area.

Graphs and Combinatorics

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors).

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal with large enough sets of numhers, there will be some well behaved fragment in the set. In Harrington's words, Ramsey-type theorems assert that complete chaos is impossible. Ramsey-type theorems have turned out to be very important in a number of branches of mathematics. In this paper we shall survey a number of basic Ramsey-type theorems, and we will then look at a selection of applications of Ramsey-type theorems and Ramsey-type ideas. In the applications we will concentrate on graph theory, logic and complexity theory. Proofs will mostly not be given in detail, but it is hoped that the reader will gain some appreciation of the usefulness and importance of the beautiful area of asymptotic combinatorics.

Journal of Combinatorial Theory, 1999

In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure. We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure.