Symmetric monochromatic subsets in colorings of the Lobachevsky plane

We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset. It follows from [B1] (see also [BP1, Theorem 1]) that for each partition of the n-dimensional space R into n pieces one of the pieces contains an unbounded centrally symmetric subset. On the other hand, R admits a partition into (n+ 1) Borel pieces containing no unbounded centrally symmetric subset. For n = 2 such a partition is drawn at the picture: b b b b b " " " " " B0 B1 B2 Taking the same partition of the Lobachevsky plane H, we can see that each cell Bi does contain a unbounded centrally symmetric subset (for such a set just take any hyperbolic line lying in Bi). We call a subset S of the hyperbolic plane H 2 centrally symmetric or else symmetric with respect to a point c ∈ H if S = fc(S) where fc : H 2 → H is the involutive isometry ofH assigning to each point x ∈ H the unique point y ∈ H such that c is the midpoint of the segment [x, y]. The map fc is called the central symmetry of H 2 with respect to the point c. The following theorem shows that the Lobachevsky plane differs dramatically from the Euclidean plane from the Ramsey point of view. Theorem 1. For any partition H = B1 ∪ · · · ∪ Bm of the Lobachevsky plane into finitely many Borel pieces one of the pieces contains an unbounded centrally symmetric subset. Proof. We shall prove a bit more: given a partition H = B1 ∪ · · · ∪ Bm of the Lobachevsky plane into m Borel pieces we shall find i ≤ m and an unbounded subset S ⊂ Bi symmetric with respect to some point c in an arbitrarily small neighborhood of some finite set F ⊂ H depending only on m. To define this set F it will be convenient to work in the Poincaré model of the Lobachevsky plane H. In this model the hyperbolic plane H is identified with 1991 Mathematics Subject Classification. 05D10, 51M09, 54H09.