{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/326362"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/326362","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Extremal problems in the cube and the grid and other combinatorial results","abstract":"This dissertation contains results from various areas of combinatorics. In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. In Chapter 2, we find the exact classification of all subsets A⊆{0,1}^n for which both A and A^c minimise the size of the neighbourhood, which answers a question of Aubrun and Szarek. Harper's inequality implies that the initial segments of the simplicial order satisfy these conditions, but we prove that in general there are non-trivial examples of such sets as well. In Chapter 3, we consider the zero-deletion shadow, which is closely related to the general coordinate deletion shadow introduced by Danh and Daykin. We prove that there is a certain order on [k]^n={0,...,k-1}^n, the n-dimensional grid of side-length k, whose initial segments minimise the size of the zero-deletion shadow. In Chapter 4, we consider the following generalisation of the Kruskal-Katona theorem on [k]^n. For a set A⊆[k]^n, define the d-shadow of A to be the set of all points x obtained from any y∈A by replacing one non-zero coordinate of y by 0. We find an order on [k]^n whose initial segments minimise the size of the d-shadow. In Chapter 5, we consider a certain combinatorial game called Toucher-Isolator game that is played on the edges of a given graph G. The value of the game on G measures how many vertices of G one of the players can achieve by using the edges claimed by her. We find the exact value of the game when G is a path or a cycle of a given length, and we prove that among the trees on n vertices, the path on n vertices has the least value of the game. These results improve previous bounds obtained by Dowden, Kang, Mikalački and Stojaković. In Chapter 6, we consider a problem in Ramsey Theory related to the Hales-Jewett theorem. We prove that for any 2-colouring of [3]^n there exists a monochromatic combinatorial line whose active coordinate set is an interval, provided that n is large. This disproves a conjecture of Conlon and Kamćev. In Chapter 7, we give a construction of a graph G that is P6-induced-saturated, where P6 is the path on 6 vertices. This answers a question of Axenovich and Csikós.","abstract_html":"This dissertation contains results from various areas of combinatorics. In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. In Chapter 2, we find the exact classification of all subsets A⊆{0,1}^n for which both A and A^c minimise the size of the neighbourhood, which answers a question of Aubrun and Szarek. Harper's inequality implies that the initial segments of the simplicial order satisfy these conditions, but we prove that in general there are non-trivial examples of such sets as well. In Chapter 3, we consider the zero-deletion shadow, which is closely related to the general coordinate deletion shadow introduced by Danh and Daykin. We prove that there is a certain order on [k]^n={0,...,k-1}^n, the n-dimensional grid of side-length k, whose initial segments minimise the size of the zero-deletion shadow. In Chapter 4, we consider the following generalisation of the Kruskal-Katona theorem on [k]^n. For a set A⊆[k]^n, define the d-shadow of A to be the set of all points x obtained from any y∈A by replacing one non-zero coordinate of y by 0. We find an order on [k]^n whose initial segments minimise the size of the d-shadow. In Chapter 5, we consider a certain combinatorial game called Toucher-Isolator game that is played on the edges of a given graph G. The value of the game on G measures how many vertices of G one of the players can achieve by using the edges claimed by her. We find the exact value of the game when G is a path or a cycle of a given length, and we prove that among the trees on n vertices, the path on n vertices has the least value of the game. These results improve previous bounds obtained by Dowden, Kang, Mikalački and Stojaković. In Chapter 6, we consider a problem in Ramsey Theory related to the Hales-Jewett theorem. We prove that for any 2-colouring of [3]^n there exists a monochromatic combinatorial line whose active coordinate set is an interval, provided that n is large. This disproves a conjecture of Conlon and Kamćev. In Chapter 7, we give a construction of a graph G that is P6-induced-saturated, where P6 is the path on 6 vertices. This answers a question of Axenovich and Csikós.","abstract_has_math":false,"creators":["Räty, Eero-Pekka"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Leader, Imre"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-04-01","date_published":"2021-04-01","updated_at":"2026-07-22T22:24:28Z","subjects":["Combinatorics","Extremal combinatorics","Discrete isoperimetric inequalities"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/dd36204d-f18b-44cc-9f28-2b9d91642ecb/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.73820","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Leader, Imre"]},{"key":"dc:creator","label":"Author","values":["Räty, Eero-Pekka"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2021-04-01"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/326362"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Combinatorics","Extremal combinatorics","Discrete isoperimetric inequalities"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/dd36204d-f18b-44cc-9f28-2b9d91642ecb/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.73820"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/440ea35d-2a36-4629-a718-f679e277e212/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This dissertation contains results from various areas of combinatorics. In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. In Chapter 2, we find the exact classification of all subsets A⊆{0,1}^n for which both A and A^c minimise the size of the neighbourhood, which answers a question of Aubrun and Szarek. Harper's inequality implies that the initial segments of the simplicial order satisfy these conditions, but we prove that in general there are non-trivial examples of such sets as well. In Chapter 3, we consider the zero-deletion shadow, which is closely related to the general coordinate deletion shadow introduced by Danh and Daykin. We prove that there is a certain order on [k]^n={0,...,k-1}^n, the n-dimensional grid of side-length k, whose initial segments minimise the size of the zero-deletion shadow. In Chapter 4, we consider the following generalisation of the Kruskal-Katona theorem on [k]^n. For a set A⊆[k]^n, define the d-shadow of A to be the set of all points x obtained from any y∈A by replacing one non-zero coordinate of y by 0. We find an order on [k]^n whose initial segments minimise the size of the d-shadow. In Chapter 5, we consider a certain combinatorial game called Toucher-Isolator game that is played on the edges of a given graph G. The value of the game on G measures how many vertices of G one of the players can achieve by using the edges claimed by her. We find the exact value of the game when G is a path or a cycle of a given length, and we prove that among the trees on n vertices, the path on n vertices has the least value of the game. These results improve previous bounds obtained by Dowden, Kang, Mikalački and Stojaković. In Chapter 6, we consider a problem in Ramsey Theory related to the Hales-Jewett theorem. We prove that for any 2-colouring of [3]^n there exists a monochromatic combinatorial line whose active coordinate set is an interval, provided that n is large. This disproves a conjecture of Conlon and Kamćev. In Chapter 7, we give a construction of a graph G that is P6-induced-saturated, where P6 is the path on 6 vertices. This answers a question of Axenovich and Csikós."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["2d2962915eeb011108eaea0d28c590e5","353adac0d1ebdfd65ab16480263c3c87"]},{"key":"dc:title","label":"Title","values":["Extremal problems in the cube and the grid and other combinatorial results"]}]}],"canonical_facts":{"dc:contributor.advisor":["Leader, Imre"],"dc:creator":["Räty, Eero-Pekka"],"dc:date.issued":["2021-04-01"],"dc:description.abstract":["This dissertation contains results from various areas of combinatorics. In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. 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We find an order on [k]^n whose initial segments minimise the size of the d-shadow. In Chapter 5, we consider a certain combinatorial game called Toucher-Isolator game that is played on the edges of a given graph G. The value of the game on G measures how many vertices of G one of the players can achieve by using the edges claimed by her. We find the exact value of the game when G is a path or a cycle of a given length, and we prove that among the trees on n vertices, the path on n vertices has the least value of the game. These results improve previous bounds obtained by Dowden, Kang, Mikalački and Stojaković. In Chapter 6, we consider a problem in Ramsey Theory related to the Hales-Jewett theorem. We prove that for any 2-colouring of [3]^n there exists a monochromatic combinatorial line whose active coordinate set is an interval, provided that n is large. This disproves a conjecture of Conlon and Kamćev. In Chapter 7, we give a construction of a graph G that is P6-induced-saturated, where P6 is the path on 6 vertices. 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