{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/91425"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/91425","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Optimal Point Sets With Few Distinct Triangles","abstract":"In this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}.","abstract_html":"In this thesis we consider the maximum number of points in <span class=\"etd-inline-math\">mathbb{R}<sup>d</sup></span> which form exactly $t$ distinct triangles, which we denote by <span class=\"etd-inline-math\">F<sub>d</sub>(t)</span>. We determine the values of <span class=\"etd-inline-math\">F<sub>d</sub>(1)</span> for all $dgeq3$, as well as determining <span class=\"etd-inline-math\">F<sub>3</sub>(2)</span>. It was known from the work of Epstein et al. cite{Epstein} that <span class=\"etd-inline-math\">F<sub>2</sub>(1) = 4</span>. Here we show somewhat surprisingly that <span class=\"etd-inline-math\">F<sub>3</sub>(1) = 4</span> and <span class=\"etd-inline-math\">F<sub>d</sub>(1) = d + 1</span>, whenever $d geq 3$, and characterize the optimal point configurations. We also show that <span class=\"etd-inline-math\">F<sub>3</sub>(2) = 6</span> and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}.","abstract_has_math":true,"creators":["Depret-Guillaume, James Serge"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Palsson, Eyvindur Ari"],"committee_members":["Senger, Steven M.","Orr, Daniel D."],"year":2019,"date_issued":"2019-07-11","date_published":"2019-07-11","updated_at":"2026-07-22T22:19:15Z","subjects":["One triangle problem","Erdos problem","Optimal configurations","Finite point configurations"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:20941"],"render_values":[{"text":"vt_gsexam:20941","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/91425","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Palsson, Eyvindur Ari"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Senger, Steven M.","Orr, Daniel D."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Depret-Guillaume, James Serge"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-07-12T08:01:18Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-07-12T08:01:18Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-07-11"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["One triangle problem","Erdos problem","Optimal configurations","Finite point configurations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:20941"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/91425"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["In this thesis we consider the following question: Given a number of triangles, t, where each of these triangles are different, we ask what is the maximum number of points that can be placed in d-dimensional space, such that every triplets of these points form the vertices of only the t allowable triangles. We answer this for every dimension, d when the number of triangles is t = 1, as well as show that when t = 2 triangle are in d = 3-dimensional space. This set of questions rises from considering the work of Erd˝os and Fishburn in higher dimensional space [EF]."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Optimal Point Sets With Few Distinct Triangles"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Palsson, Eyvindur Ari"],"dc:contributor.committeemember":["Senger, Steven M.","Orr, Daniel D."],"dc:contributor.department":["Mathematics"],"dc:creator":["Depret-Guillaume, James Serge"],"dc:date.accessioned":["2019-07-12T08:01:18Z"],"dc:date.available":["2019-07-12T08:01:18Z"],"dc:date.issued":["2019-07-11"],"dc:description.abstract":["In this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}."],"dc:description.abstractgeneral":["In this thesis we consider the following question: Given a number of triangles, t, where each of these triangles are different, we ask what is the maximum number of points that can be placed in d-dimensional space, such that every triplets of these points form the vertices of only the t allowable triangles. We answer this for every dimension, d when the number of triangles is t = 1, as well as show that when t = 2 triangle are in d = 3-dimensional space. This set of questions rises from considering the work of Erd˝os and Fishburn in higher dimensional space [EF]."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:20941"],"dc:identifier.uri":["http://hdl.handle.net/10919/91425"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["One triangle problem","Erdos problem","Optimal configurations","Finite point configurations"],"dc:title":["Optimal Point Sets With Few Distinct Triangles"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:15Z"}