{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T23:46:38Z","timestamp":1768779998086,"version":"3.49.0"},"reference-count":50,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"name":"the National Science Foundation","award":["DMS-1600781"],"award-info":[{"award-number":["DMS-1600781"]}]},{"name":"the National Science Foundation","award":["DMS-1901753"],"award-info":[{"award-number":["DMS-1901753"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:p> Analogues of Ramsey\u2019s 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\u2019s recent result for the triangle-free Henson graph, we prove that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey\u2019s 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. <\/jats:p>","DOI":"10.1142\/s0219061322500180","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T09:18:28Z","timestamp":1655457508000},"source":"Crossref","is-referenced-by-count":8,"title":["The Ramsey theory of Henson graphs"],"prefix":"10.1142","volume":"23","author":[{"given":"Natasha","family":"Dobrinen","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Notre Dame, 255 Hurley Bldg Notre Dame, IN 46556 USA"}]}],"member":"219","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"issue":"3","key":"S0219061322500180BIB001","doi-asserted-by":"crossref","first-page":"572","DOI":"10.2307\/2273534","volume":"43","author":"Abramson F. G.","year":"1978","journal-title":"J. Symb. 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