{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/34320"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/34320","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Forbidden substructures: induced subgraphs, Ramsey games, and sparse hypergraphs","abstract":"We study problems in extremal combinatorics with respect to forbidden induced subgraphs, forbidden colored subgraphs, and forbidden subgraphs. In Chapter 2, we determine exactly which graphs H have the property that almost every H-free graph has a vertex partition into k cliques and independent sets and provide a characterization. Such graphs contain homogeneous sets of size linear in the number of vertices, and so this result provides a strong partial result toward proving the Erdos-Hajnal conjecture. In Chapter 3, we study a Ramsey-type game in an online and random setting. The player must color edges of K_n in an order chosen uniformly at random, and loses when she has created a monochromatic triangle. We provide upper bounds on the threshold for the number of edges the player is almost surely able to paint before losing in the k-color game. When k > 2, these upper bounds provide the first separation from the online threshold. In Chapter 4, we consider the family of 3-uniform hypergraphs that do not contain a copy of F_5, sometimes called the generalized triangle. We extend known extremal results to the sparse random setting, proving that with probability tending to 1 the largest subgraph of the random 3-uniform hypergraph that does not contain F_5 is tripartite.","abstract_html":"We study problems in extremal combinatorics with respect to forbidden induced subgraphs, forbidden colored subgraphs, and forbidden subgraphs. In Chapter 2, we determine exactly which graphs H have the property that almost every H-free graph has a vertex partition into k cliques and independent sets and provide a characterization. Such graphs contain homogeneous sets of size linear in the number of vertices, and so this result provides a strong partial result toward proving the Erdos-Hajnal conjecture. In Chapter 3, we study a Ramsey-type game in an online and random setting. The player must color edges of K_n in an order chosen uniformly at random, and loses when she has created a monochromatic triangle. We provide upper bounds on the threshold for the number of edges the player is almost surely able to paint before losing in the k-color game. When k > 2, these upper bounds provide the first separation from the online threshold. In Chapter 4, we consider the family of 3-uniform hypergraphs that do not contain a copy of F_5, sometimes called the generalized triangle. We extend known extremal results to the sparse random setting, proving that with probability tending to 1 the largest subgraph of the random 3-uniform hypergraph that does not contain F_5 is tripartite.","abstract_has_math":false,"creators":["Butterfield, Jane"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Balogh, József","Kostochka, Alexandr V.","West, Douglas B.","Lidicky, Bernard"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-09-18T21:11:13Z","date_published":"2012-09-18T21:11:13Z","updated_at":"2026-07-22T22:25:31Z","subjects":["induced subgraph","Ramsey theory","extremal combinatorics"],"languages":["en"],"rights":["Copyright 2012 Jane Butterfield"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/34320","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Balogh, József","Kostochka, Alexandr V.","West, Douglas B.","Lidicky, Bernard"]},{"key":"dc:creator","label":"Author","values":["Butterfield, Jane"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-09-18T21:11:13Z","2012-08"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["induced subgraph","Ramsey theory","extremal combinatorics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Jane Butterfield"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/34320"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We study problems in extremal combinatorics with respect to forbidden induced subgraphs, forbidden colored subgraphs, and forbidden subgraphs. In Chapter 2, we determine exactly which graphs H have the property that almost every H-free graph has a vertex partition into k cliques and independent sets and provide a characterization. Such graphs contain homogeneous sets of size linear in the number of vertices, and so this result provides a strong partial result toward proving the Erdos-Hajnal conjecture. In Chapter 3, we study a Ramsey-type game in an online and random setting. The player must color edges of K_n in an order chosen uniformly at random, and loses when she has created a monochromatic triangle. We provide upper bounds on the threshold for the number of edges the player is almost surely able to paint before losing in the k-color game. When k > 2, these upper bounds provide the first separation from the online threshold. In Chapter 4, we consider the family of 3-uniform hypergraphs that do not contain a copy of F_5, sometimes called the generalized triangle. We extend known extremal results to the sparse random setting, proving that with probability tending to 1 the largest subgraph of the random 3-uniform hypergraph that does not contain F_5 is tripartite.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-06T19:38:09Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Butterfield_Jane.pdf: 602646 bytes, checksum: f1612ece7e2be057a2e2c6db56ae9841 (MD5)","Made available in DSpace on 2012-09-18T21:11:13Z (GMT). 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Such graphs contain homogeneous sets of size linear in the number of vertices, and so this result provides a strong partial result toward proving the Erdos-Hajnal conjecture. In Chapter 3, we study a Ramsey-type game in an online and random setting. The player must color edges of K_n in an order chosen uniformly at random, and loses when she has created a monochromatic triangle. We provide upper bounds on the threshold for the number of edges the player is almost surely able to paint before losing in the k-color game. When k > 2, these upper bounds provide the first separation from the online threshold. In Chapter 4, we consider the family of 3-uniform hypergraphs that do not contain a copy of F_5, sometimes called the generalized triangle. We extend known extremal results to the sparse random setting, proving that with probability tending to 1 the largest subgraph of the random 3-uniform hypergraph that does not contain F_5 is tripartite.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-06T19:38:09Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Butterfield_Jane.pdf: 602646 bytes, checksum: f1612ece7e2be057a2e2c6db56ae9841 (MD5)","Made available in DSpace on 2012-09-18T21:11:13Z (GMT). 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