39 research outputs found
A combinatorial proof of the extension property for partial isometries
We present a short and self-contained proof of the extension property for
partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes
Mathematicae Universitatis Carolina
Big Ramsey degrees using parameter spaces
We show that the universal homogeneous partial order has finite big Ramsey
degrees and discuss several corollaries. Our proof uses parameter spaces and
the Carlson-Simpson theorem rather than (a strengthening of) the
Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary
tools used to give bounds on big Ramsey degrees elsewhere (originating from
work of Laver and Milliken).
This new technique has many additional applications. To demonstrate this, we
show that the homogeneous universal triangle-free graph has finite big Ramsey
degrees, thus giving a short proof of a recent result of Dobrinen.Comment: 19 pages, 2 figure