Indivisibility and Alpha-morphisms

A relation R is p-divisible if for any partition of its basis into p + 1 subsets, R is embedded into the union of p of them. We prove that any countable p-divisible relation embeds two copies of itself intersecting in at most p− 1 elements. A relation R is indivisible if for any partition of its set of vertices into two subsets V1 and V2, there exists i ∈ {1, 2} such that R is embedded into R(Vi), where R(Vi) denotes the restriction of R to Vi. Clearly, there is no finite indivisible relation with more than one vertex. In the infinite case, in addition to some trivial examples of indivisible relations such that complete graphs or the order type of natural numbers, one can easily check that Rado’s graph and the order type of the rationals are also indivisible. The starting point of this work is the following conjecture of Fräıssé : if R is indivisible, there exist two disjoint embeddings from R into itself. This conjecture has been proved true in the countable case by Pouzet [6] : a countable counterexample would give a non-principal analytic ultrafilter on ω, contrary to a result of Sierpinski. For recent developments on Fräıssé’s conjecture in the general case, see Bishop’s work on ultrafilters [1]. One possible generalization of indivisibility is to increase the number of blocks of the partition, thus, a relation is p-divisible if for any partition of its basis into p+ 1 subsets {V1, V2, . . . , Vp+1} there exists i such that R is embedded into R \ Vi (clearly, 1-divisible is indivisible). Answering a question of Hajnal, Laver proved in [5] that for any countable linear ordering L there exists a p such that L is p-divisible. We prove in this paper a generalization of Pouzet’s theorem : if R is a countable p-divisible relation, one can find two copies of R into itself intersecting in at most p− 1 vertices (p = 1 gives Pouzet’s theorem). We give a purely combinatorial proof of this theorem which is based on the notion of α-morphism (the key idea of Ehrenfeucht-Fräıssé games). A 0-morphism from R into R′ is a local isomorphism (an isomorphism from a finite restriction of R into a finite restriction of R′). An α + 1-morphism is a local isomorphism from R into R′ which can be extended on any finite subset of R as an α-morphism. Lastly, if α is a limit ordinal, an α-morphism is a β-morphism for any β < α. According to this definition, a relation R α-embeds a relation R′ if the empty morphism is an α-morphism from R′ into R. The fundamental theorem of Fräıssé asserts that if R and R′ are countable and R ω1-embeds R′ then R embeds R′. In this paper, we use this notion in order to construct inductively the disjoint (resp. nearly disjoint) copies of an indivisible (resp. p-divisible) relation. Here are the two steps of the proof, when R is an indivisible relation : First we prove that if for any countable α, one can find two disjoint subsets V α 1 and V α 2 such that R is α-embedded into R(V α 1 ) and into R(V α 2 ), then R embeds two disjoint copies of itself. We then prove that if R is indivisible, one can always find such subsets V α 1 and V α 2 for any countable α.