{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T00:31:20Z","timestamp":1773966680286,"version":"3.50.1"},"reference-count":19,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","funder":[{"name":"J. A. Grochow\u2019s NSF","award":["CISE-2047756"],"award-info":[{"award-number":["CISE-2047756"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p> In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Prior to our paper, only [Formula: see text] and [Formula: see text] were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest. <\/jats:p><jats:p> We complement our upper bounds with some lower bounds. Namely, we show that any square representation of [Formula: see text] requires at least 14 points, any square representation of [Formula: see text] requires at least 11 points, and any square representation of [Formula: see text] requires at least 12 points. Our technique adapts previous work of Alm et\u00a0al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom. <\/jats:p>","DOI":"10.1142\/s0218196723500595","type":"journal-article","created":{"date-parts":[[2023,8,19]],"date-time":"2023-08-19T05:28:20Z","timestamp":1692422900000},"page":"1571-1598","source":"Crossref","is-referenced-by-count":3,"title":["Directed Ramsey and anti-Ramsey schemes and the Flexible Atom Conjecture"],"prefix":"10.1142","volume":"33","author":[{"given":"Jeremy F.","family":"Alm","sequence":"first","affiliation":[{"name":"Department of Mathematics, Lamar University, Beaumont, TX 77710, USA"}]},{"given":"Michael","family":"Levet","sequence":"additional","affiliation":[{"name":"Department of Computer Science, College of Charleston, Charleston, SC 29492, USA"}]}],"member":"219","published-online":{"date-parts":[[2023,9,21]]},"reference":[{"key":"S0218196723500595BIB001","doi-asserted-by":"publisher","DOI":"10.37236\/773"},{"key":"S0218196723500595BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-022-00791-4"},{"key":"S0218196723500595BIB003","first-page":"17.8.4","volume":"20","author":"Alm J. 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