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A note on diameter-Ramsey sets
A finite set AâRd is called diameter-Ramsey if for every râN, there exists some nâN and a finite set BâRn with diam(A)=diam(B) such that whenever B is coloured with r colours, there is a monochromatic set AâČâB which is congruent to A. We prove that sets of diameter 1 with circumradius larger than 1/2ââ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than 135â are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and R\"odl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
revisio
The critical window for the classical Ramsey-Tur\'an problem
The first application of Szemer\'edi's powerful regularity method was the
following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any
K_4-free graph on N vertices with independence number o(N) has at most (1/8 +
o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising
geometric construction, utilizing the isoperimetric inequality for the high
dimensional sphere, of a K_4-free graph on N vertices with independence number
o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in
1976, several problems have been asked on estimating the minimum possible
independence number in the critical window, when the number of edges is about
N^2 / 8. These problems have received considerable attention and remained one
of the main open problems in this area. In this paper, we give nearly
best-possible bounds, solving the various open problems concerning this
critical window.Comment: 34 page
Lines in Euclidean Ramsey theory
Let be a sequence of points on a line with consecutive points of
distance one. For every natural number , we prove the existence of a
red/blue-coloring of containing no red copy of and no
blue copy of for any . This is best possible up to the
constant in the exponent. It also answers a question of Erd\H{o}s, Graham,
Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every
natural number , there is a set and a
red/blue-coloring of containing no red copy of and no
blue copy of .Comment: 7 page
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