SYMMETRIZED INDUCED RAMSEY THEOREMS STEFAN GESCHKE AND MENACHEM KOJMAN Abstract. We prove induced Ramsey theorems in which the induced monochromatic subgraph satisfies that some of its partial automor- phisms extend to automorphisms of the colored graph. 1. Introduction By “graph” we always mean “simple graph”. 1.1. Symmetrized subgraph relations. A partial automorphism of a graph H is an isomorphism between two induced subgraphs of H . Sup- pose H is an induced subgraph of G and f is a partial automorphism of H . We say that f is implemented in G if there exists f ∗ ∈ Aut(G) so that f ⊆ f ∗ . For a set F of partial automorphisms of H , the relation symbol H ⊆ F G denotes that H is an induced subgraph of G and every f ∈ F is implemented in G. Let H ≤ F G mean that there exists some H ′ ⊆ F G such that H ′ is isomorphic to H . Let P = P (H ) denote the set of all partial finite automorphisms of H , let P 1 = P 1 (H ) denote the set of all partial automorphisms of size 1 (that is, the set of all maps whose domain and range are singleton subsets of V (H )) and let A = A(H ) abbreviate Aut(H ). Thus, H ⊆ P G means that every finite partial automorphism of H extends to a total automorphism of G, H ⊆ P 1 G means that every vertex of H can be moved to the every vertex of H by an automorphism of G and H ⊆ A G means that every automorphism of H extends to an automorphism of G. If H ⊆ A G, we say that H is a symmetrized induced subgraph of G. Thus, H ≤ A G iff G contains a symmetrized copy of H . Let Γ denote Rado’s countable and universal random graph. Every iso- morphism between two finite induced subgraphs of Γ is implemented in Γ, so every induced subgraph G of Γ satisfies G ⊆ P Γ. As every countable graph embeds as an induced subgraph into Γ, it holds that for every countable graph G, (1) G ≤ P Γ. 2000 Mathematics Subject Classification. Primary: 05C15; Secondary: 05C55, 05C80. Key words and phrases. Ramsey Theory, partial automorphism, random graph. The research for this paper was supported by G.I.F. Research Grant No. I-802- 195.6/2003. 1