Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $k\ge 4$, the
$k$-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.

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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: acc-phys