## On a Ramsey-type topological theorem

A topological space is left separated if there is a well-order < on X and a nbd U(x) for each x in X such that min U(x)=x for each x in X.

Theorem If X is regular and left-separated, then we can colour the pairs of X with 2 colours such that any homogeneous set is discreet.

A certain converse was proved recently by S.Todorcevic and W.Weiss:

Theorem (S.Todorcevic-W.Weiss) If the r-tuples of a not left separated metric space are coloured with n\in omega colours then there exists a not discreet homogeneous subset.

With the notation of the partition calculus, this can be expressed as

If X is a not left separated metric space then for any 2 <= r, n < omega we have X --> (omega +1)_n^r

We extended this result for first countable monotonically normal spaces. (A T_1-space is monotonically normal if to each pair (x,U), where x\in X and U is an open nbd of x, we can assign an open set U' such that x \in U' and for any two pairs (x,U),(y,V), if x\not\in V and y\not\in U then U'\cap V'=\emptyset. Any metrizable and any ordered space is monotonically normal.)

The following property is connected with this subject:

A space X is said to be weakly separated if there exist a a nbd-assignment U(x) (x\in X) such that any infinite A\subset X contains a sequence \{ x_n: n\in \omega \} with x_n \not\in U(x_m) for n<m<\omega .

Theorem If X is regular and weakly left separated, then we can colour the pairs of X with 3(!) colours such that any homogeneous set is discreet.

Theorem A monotonically normal space X with countable tightness is weakly left separated iff it is left separated in type |X|.

Reference

S.Todorcevic--W.Weiss: Partitioning Metric Spaces, manuscript, September 1995, 1-9.