433 research outputs found
Unordered Canonical Ramsey Numbers
AbstractWe define a weak form of canonical colouring, based on that of P. Erdős and R. Rado (1950, J. London Math. Soc.25, 249–255). This yields a class of unordered canonical Ramsey numbers CR(s, t), again related to the canonical Ramsey numbers ER(2; s) of Erdős and Rado. We present upper and lower bounds (the latter via a construction) for CR(s, t) which are significantly tighter than the best-known corresponding bounds for ER(2; s)
On union ultrafilters
We present some new results on union ultrafilters. We characterize stability
for union ultrafilters and, as the main result, we construct a new kind of
unordered union ultrafilter
On the hard sphere model and sphere packings in high dimensions
We prove a lower bound on the entropy of sphere packings of of
density . The entropy measures how plentiful such
packings are, and our result is significantly stronger than the trivial lower
bound that can be obtained from the mere existence of a dense packing. Our
method also provides a new, statistical-physics-based proof of the lower bound on the maximum sphere packing density by showing
that the expected packing density of a random configuration from the hard
sphere model is at least when the
ratio of the fugacity parameter to the volume covered by a single sphere is at
least . Such a bound on the sphere packing density was first achieved
by Rogers, with subsequent improvements to the leading constant by Davenport
and Rogers, Ball, Vance, and Venkatesh
Permutations on the random permutation
The random permutation is the Fra\"iss\'e limit of the class of finite
structures with two linear orders. Answering a problem stated by Peter Cameron
in 2002, we use a recent Ramsey-theoretic technique to show that there exist
precisely 39 closed supergroups of the automorphism group of the random
permutation, and thereby expose all symmetries of this structure. Equivalently,
we classify all structures which have a first-order definition in the random
permutation.Comment: 18 page
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