{"id":{"repo_id":"toronto-retro","oai_identifier":"oai:utoronto.scholaris.ca:1807/68124"},"canonical_url":"https://search.dev.ndltd.org/etd/toronto-retro/oai:utoronto.scholaris.ca:1807/68124","repository":{"repo_id":"toronto-retro","name":"University of Toronto","base_url":"https://utoronto.scholaris.ca/server/oai/request"},"display":{"title":"A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees","abstract":"Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Then non-(2 <κ)-special tree → (κ + ξ)(2/ k). This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2 <κ)+, the simplest example of a non-(2 <κ)-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Let P be a partially ordered set such that P → (2<κ)(1/2<κ). Then P → (κ + ξ)(2/k).","abstract_html":"Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| &lt; κ, and let k be any natural number. Then non-(2 &lt;κ)-special tree → (κ + ξ)(2/ k). This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2 &lt;κ)+, the simplest example of a non-(2 &lt;κ)-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| &lt; κ, and let k be any natural number. Let P be a partially ordered set such that P → (2&lt;κ)(1/2&lt;κ). Then P → (κ + ξ)(2/k).","abstract_has_math":false,"creators":["Brodsky, Meir Ari"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Todorcevic, B Stevo"],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-06","date_published":"2014-06","updated_at":"2026-07-27T21:28:01Z","subjects":["elementary submodels","nonreflecting ideals","nonspecial trees","partial orders","partition rrelations","stationary subtrees"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1807/68124","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Todorcevic, B Stevo"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Brodsky, Meir Ari"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-04-17T15:08:16Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-04-17T15:08:16Z"]},{"key":"dc:date.issued","label":"Date","values":["2014-06"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["elementary submodels","nonreflecting ideals","nonspecial trees","partial orders","partition rrelations","stationary subtrees"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1807/68124"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Then non-(2 <κ)-special tree → (κ + ξ)(2/ k). This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2 <κ)+, the simplest example of a non-(2 <κ)-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Let P be a partially ordered set such that P → (2<κ)(1/2<κ). Then P → (κ + ξ)(2/k)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees"]}]}],"canonical_facts":{"dc:contributor.advisor":["Todorcevic, B Stevo"],"dc:contributor.department":["Mathematics"],"dc:creator":["Brodsky, Meir Ari"],"dc:date.accessioned":["2015-04-17T15:08:16Z"],"dc:date.available":["2015-04-17T15:08:16Z"],"dc:date.issued":["2014-06"],"dc:description.abstract":["Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Then non-(2 <κ)-special tree → (κ + ξ)(2/ k). This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal (2 <κ)+, the simplest example of a non-(2 <κ)-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let κ be any infinite regular cardinal, let ξ be any ordinal such that 2|ξ| < κ, and let k be any natural number. Let P be a partially ordered set such that P → (2<κ)(1/2<κ). Then P → (κ + ξ)(2/k)."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1807/68124"],"dc:subject":["elementary submodels","nonreflecting ideals","nonspecial trees","partial orders","partition rrelations","stationary subtrees"],"dc:title":["A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T21:28:01Z"}