91,224 research outputs found
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
The Exchange Value Embedded In A Transport System
This paper shows that a well designed transport system has an embedded
exchange value by serving as a market for potential exchange between consumers.
Under suitable conditions, one can improve the welfare of consumers in the
system simply by allowing some exchange of goods between consumers during
transportation without incurring additional transportation costs. We propose an
explicit valuation formula to measure this exchange value for a given
compatible transport system. This value is always nonnegative and bounded from
above. Criteria based on transport structures, preferences and prices are
provided to determine the existence of a positive exchange value. Finally, we
study a new optimal transport problem with an objective taking into account of
both transportation cost and exchange value.Comment: 20 pages, 6 figure
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