On Ramsey properties of classes with forbidden trees

Let A, B be digraphs and r an integer. Then there exists a digraph C such that whenever the copies of A in C are coloured with r colours, then there exists a copy B of B in C such that all the copies of A in B have the same colour. It is, however, not hard to show that this statement is false: Let A be a single arc and B the directed 4-cycle. Given any digraph C, the adversary can colour the arcs of C with r ≥ 2 colours as follows: First, fix an arbitrary linear ordering of the vertex set of C; then colour every arc of C “red” if it goes “forward” with respect to the ordering on its endpoints, and “blue” if it goes “backward”. Now, no matter how we order the vertices of B it will contain both a forward and a backward arc -thus no copy of B in C can have all its arcs in the same colour class.

This issue can be fixed by considering ordered structures: in this case we would consider digraphs with an additional linear ordering of its vertices. Then a “forward” arc and a “backward” arc are distinct, non-isomorphic ordered digraphs and, in fact, the above statement becomes true.

Theorem 1.1 (Nešetřil-Rödl [22]). Let A, B be ordered digraphs and r an integer. Then there exists an ordered digraph C such that whenever the copies of A in C are coloured with

The aim of this paper is to prove analogous results for A, B, C belonging to specific classes of structures. Perhaps the simplest example of such a result is the analogue of Theorem 1.1 where we replace “ordered digraphs” with “ordered K n -free undirected graphs”, proved by Folkman [11] for A = K 2 , B = K n-1 , as well as for A = K 1 and any K n -free B, and by Nešetřil and Rödl [22,25] in general. Here we study classes of ordered digraphs (and, more generally, relational structures) obtained not by forbidding one subgraph, such as the complete graph in the example above, but by forbidding all homomorphic images of a given set F of oriented trees. In this context a homomorphism is a mapping that preserves the arcs but it need not preserve the linear ordering.

Thus for a (possibly infinite) set F of oriented trees, let Forb h (F) be the class of all ordered digraphs that admit no homomorphism from any tree in F. These classes are interesting in the context of constraint satisfaction problems. For a finite digraph H, CSP(H) denotes the class of all digraphs that admit a homomorphism to H. The case where CSP(H) = Forb h (F) for a set F of trees corresponds to constraint satisfaction problems with tree duality (also known as width-one constraint satisfaction problems). Ramsey theory provides a way to recognise digraphs H that define CSPs with tree duality (see Section 8). Now, however, another issue arises that can be illustrated with this example: Let P 3 be the directed path with three arcs and consider C = Forb h (F) for F = {P 3 }. For A = P 1 , an arc ordered forward, and B = P 2 , the directed path with two arcs 0 → 1 → 2 ordered 0 ≺ 1 ≺ 2, there can be no P 3 -free C with the Ramsey property for A and B: In any C we can colour an arc “red” if there is another arc going out from its head, and “blue” otherwise. In any copy of B = P 2 in C the first arc will be red; if there is a monochromatic copy B of B, then its second arc must also be coloured red. This implies, however, that there is a homomorphic image of P 3 in C.

The way to tackle this problem is to introduce new unary relations on the vertices in a clever way (determined by the trees in F). In our example, we would impose a unary relation on a vertex v (let us call the unary relation “square”) whenever there is an arc leaving v, and another unary relation (“circle”) whenever there is a copy of P 2 leaving v. Then there is always both a square and a circle on the starting vertex of P 2 , but never a circle on its middle vertex. Hence the two arcs of P 2 are no longer isomorphic induced subgraphs and we never colour them both: A cannot be an arc both with a circle on its tail and without one.

The precise fashion in which the unary relations are introduced is described in Section 4. Not always is it possible to use only finitely many unary relations. Interestingly, it turns out that a finite number of unary relations suffice if and only if Forb h (F) = CSP(H) for some finite H.

The main result of this paper is the Ramsey property of any class Forb h (F) of ordered relational structures expanded by a number of unary relations, with F being a set of relational trees. The setting is properly defined in Section 2; the unary relations are introduced in Section 4. Section 5 presents the main result, which is then proved in Sections 6 and 7. Section 8 describes a link between constraint satisfaction problems with tree duality and our Ramsey classes; the paper then concludes with a number of final comments.

ON RAMSEY PROPERTIES OF CLASSES WITH FO