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Strong Ramsey Games in Unbounded Time
For two graphs and the strong Ramsey game on the
board and with target is played as follows. Two players alternately
claim edges of . The first player to build a copy of wins. If none of
the players win, the game is declared a draw. A notorious open question of Beck
asks whether the first player has a winning strategy in
in bounded time as . Surprisingly, in a recent paper Hefetz
et al. constructed a -uniform hypergraph for which they proved
that the first player does not have a winning strategy in
in bounded time. They naturally ask
whether the same result holds for graphs. In this paper we make further
progress in decreasing the rank.
In our first result, we construct a graph (in fact )
and prove that the first player does not have a winning strategy in
in bounded time. As an application of this
result we deduce our second result in which we construct a -uniform
hypergraph and prove that the first player does not have a winning
strategy in in bounded time. This improves the
result in the paper above.
An equivalent formulation of our first result is that the game
is a draw. Another reason for interest
on the board is a folklore result that the disjoint
union of two finite positional games both of which are first player wins is
also a first player win. An amusing corollary of our first result is that at
least one of the following two natural statements is false: (1) for every graph
, is a first player win; (2) for every graph
if is a first player win, then
is also a first player win.Comment: 18 pages, 46 figures; changes: fully reworked presentatio
Strong Ramsey Games in Unbounded Time
For two graphs and the strong Ramsey game on the
board and with target is played as follows. Two players alternately
claim edges of . The first player to build a copy of wins. If none of
the players win, the game is declared a draw. A notorious open question of Beck
asks whether the first player has a winning strategy in
in bounded time as . Surprisingly, in a recent paper Hefetz
et al. constructed a -uniform hypergraph for which they proved
that the first player does not have a winning strategy in
in bounded time. They naturally ask
whether the same result holds for graphs. In this paper we make further
progress in decreasing the rank.
In our first result, we construct a graph (in fact )
and prove that the first player does not have a winning strategy in
in bounded time. As an application of this
result we deduce our second result in which we construct a -uniform
hypergraph and prove that the first player does not have a winning
strategy in in bounded time. This improves the
result in the paper above.
An equivalent formulation of our first result is that the game
is a draw. Another reason for interest
on the board is a folklore result that the disjoint
union of two finite positional games both of which are first player wins is
also a first player win. An amusing corollary of our first result is that at
least one of the following two natural statements is false: (1) for every graph
, is a first player win; (2) for every graph
if is a first player win, then
is also a first player win.Comment: 17 pages, 48 figures; improved presentation, particularly in section
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
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