{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/29436"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/29436","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Problems in extremal combinatorics","abstract":"We consider a variety of problems in extremal graph and set theory. Given a property $\\Gamma$ and a family of sets ${\\mathcal F}$, let $f({\\mathcal F},\\Gamma)$ be the size of the largest subfamily of ${\\mathcal F}$ having property $\\Gamma$. Let $f(m,\\Gamma)$ be the minimum of $f({\\mathcal F},\\Gamma)$ over all families of size $m$ where $m$ is a positive integer. A family $\\mathcal{F}$ is {\\it $B_d$-free} if it has no subfamily $\\mathcal{F}'=\\{F_I: I \\subseteq [d]\\}$ of $2^d$ distinct sets such that for every $I,J \\subseteq [d]$, both $F_I \\cup F_J=F_{I \\cup J}$ and $F_I \\cap F_J = F_{I \\cap J}$ hold. A family $\\mathcal{F}$ is $a$-{\\it union-free} if $F_1\\cup \\dots \\cup F_a \\neq F_{a+1}$ whenever $F_1,\\dots,F_{a+1}$ are distinct sets in $\\mathcal{F}$. We prove a conjecture of Erd\\H os and Shelah that $f(m, B_2\\text{\\rm -free})=\\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\\text{\\rm -free})$ and $f(m,a\\text{\\rm -union-free})$. A graph $G$ is {\\it $F$-saturated } if it does not contain $F$ as a subgraph but the addition of any new edge creates at least one copy of $F$ in $G$. We focus on finding the minimum size of an $n$-vertex $F$-saturated graph, denoted by $\\sat(n,F)$. We prove $ \\sat(n,C_k) = n + \\frac{n}{k} + O((\\frac{n}{k^2}) + k^2)$ for all $n\\geq k\\geq 3$, where $C_k$ is a cycle with length $k$. We conjecture that our three constructions are optimal. We obtain the exact asymptotics for the number of $n$-vertex graphs of diameter $d$, extending earlier results to hold for almost all $d$ and $n$. Additionally, we find the typical structure of almost all $n$-vertex graphs with diameter of at least $d$. In the case $d < n - c_1 \\log n$, the typical graph of diameter $d$ consists of an induced path of length $d$ and a highly connected block of order $n-d+3$. In the case $d > n - c_2 \\log n$, the typical graph has a completely different snake-like structure. We also extend the results to random graphs of diameter $d$ with edge probability $p$.","abstract_html":"We consider a variety of problems in extremal graph and set theory. Given a property $\\Gamma$ and a family of sets ${\\mathcal F}$, let $f({\\mathcal F},\\Gamma)$ be the size of the largest subfamily of ${\\mathcal F}$ having property $\\Gamma$. Let $f(m,\\Gamma)$ be the minimum of $f({\\mathcal F},\\Gamma)$ over all families of size $m$ where $m$ is a positive integer. A family $\\mathcal{F}$ is {\\it <span class=\"etd-inline-math\">B<sub>d</sub></span>-free} if it has no subfamily <span class=\"etd-inline-math\">\\mathcal{F}&#x27;=\\{F<sub>I</sub>: I \\subseteq [d]\\}</span> of <span class=\"etd-inline-math\">2<sup>d</sup></span> distinct sets such that for every $I,J \\subseteq [d]$, both <span class=\"etd-inline-math\">F<sub>I</sub> \\cup F<sub>J</sub>=F<sub>I \\cup J</sub></span> and <span class=\"etd-inline-math\">F<sub>I</sub> \\cap F<sub>J</sub> = F<sub>I \\cap J</sub></span> hold. A family $\\mathcal{F}$ is $a$-{\\it union-free} if <span class=\"etd-inline-math\">F<sub>1</sub>\\cup \\dots \\cup F<sub>a</sub> \\neq F<sub>a+1</sub></span> whenever <span class=\"etd-inline-math\">F<sub>1</sub>,\\dots,F<sub>a+1</sub></span> are distinct sets in $\\mathcal{F}$. We prove a conjecture of Erd\\H os and Shelah that <span class=\"etd-inline-math\">f(m, B<sub>2</sub>\\text{\\rm -free})=\\Theta(m<sup>2/3</sup>)</span>. We also obtain lower and upper bounds for <span class=\"etd-inline-math\">f(m, B<sub>d</sub>\\text{\\rm -free})</span> and $f(m,a\\text{\\rm -union-free})$. A graph $G$ is {\\it $F$-saturated } if it does not contain $F$ as a subgraph but the addition of any new edge creates at least one copy of $F$ in $G$. We focus on finding the minimum size of an $n$-vertex $F$-saturated graph, denoted by $\\sat(n,F)$. We prove <span class=\"etd-inline-math\"> \\sat(n,C<sub>k</sub>) = n + \\frac{n}{k} + O((\\frac{n}{k<sup>2</sup>}) + k<sup>2</sup>)</span> for all $n\\geq k\\geq 3$, where <span class=\"etd-inline-math\">C<sub>k</sub></span> is a cycle with length $k$. We conjecture that our three constructions are optimal. We obtain the exact asymptotics for the number of $n$-vertex graphs of diameter $d$, extending earlier results to hold for almost all $d$ and $n$. Additionally, we find the typical structure of almost all $n$-vertex graphs with diameter of at least $d$. In the case <span class=\"etd-inline-math\">d &lt; n - c<sub>1</sub> \\log n</span>, the typical graph of diameter $d$ consists of an induced path of length $d$ and a highly connected block of order $n-d+3$. In the case <span class=\"etd-inline-math\">d &gt; n - c<sub>2</sub> \\log n</span>, the typical graph has a completely different snake-like structure. We also extend the results to random graphs of diameter $d$ with edge probability $p$.","abstract_has_math":true,"creators":["Kim, Youn-Jin"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Furedi, Zoltan","Kostochka, Alexandr V.","West, Douglas B.","Balogh, József"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-02-01T00:46:19Z","date_published":"2012-02-01T00:46:19Z","updated_at":"2026-07-22T22:25:27Z","subjects":["graphs","cycles","extremal graphs","minimal saturated graphs","diameter","random graphs","set families","boolean algebras"],"languages":["en"],"rights":["Copyright 2011 Younjin Kim"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/29436","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Furedi, Zoltan","Kostochka, Alexandr V.","West, Douglas B.","Balogh, József"]},{"key":"dc:creator","label":"Author","values":["Kim, Youn-Jin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-02-01T00:46:19Z","2014-02-01T11:00:26Z","2011-12"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation / Thesis","text"]},{"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":["graphs","cycles","extremal graphs","minimal saturated graphs","diameter","random graphs","set families","boolean algebras"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Younjin Kim"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/29436"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider a variety of problems in extremal graph and set theory. Given a property $\\Gamma$ and a family of sets ${\\mathcal F}$, let $f({\\mathcal F},\\Gamma)$ be the size of the largest subfamily of ${\\mathcal F}$ having property $\\Gamma$. Let $f(m,\\Gamma)$ be the minimum of $f({\\mathcal F},\\Gamma)$ over all families of size $m$ where $m$ is a positive integer. A family $\\mathcal{F}$ is {\\it $B_d$-free} if it has no subfamily $\\mathcal{F}'=\\{F_I: I \\subseteq [d]\\}$ of $2^d$ distinct sets such that for every $I,J \\subseteq [d]$, both $F_I \\cup F_J=F_{I \\cup J}$ and $F_I \\cap F_J = F_{I \\cap J}$ hold. A family $\\mathcal{F}$ is $a$-{\\it union-free} if $F_1\\cup \\dots \\cup F_a \\neq F_{a+1}$ whenever $F_1,\\dots,F_{a+1}$ are distinct sets in $\\mathcal{F}$. We prove a conjecture of Erd\\H os and Shelah that $f(m, B_2\\text{\\rm -free})=\\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\\text{\\rm -free})$ and $f(m,a\\text{\\rm -union-free})$. A graph $G$ is {\\it $F$-saturated } if it does not contain $F$ as a subgraph but the addition of any new edge creates at least one copy of $F$ in $G$. We focus on finding the minimum size of an $n$-vertex $F$-saturated graph, denoted by $\\sat(n,F)$. We prove $ \\sat(n,C_k) = n + \\frac{n}{k} + O((\\frac{n}{k^2}) + k^2)$ for all $n\\geq k\\geq 3$, where $C_k$ is a cycle with length $k$. We conjecture that our three constructions are optimal. We obtain the exact asymptotics for the number of $n$-vertex graphs of diameter $d$, extending earlier results to hold for almost all $d$ and $n$. Additionally, we find the typical structure of almost all $n$-vertex graphs with diameter of at least $d$. In the case $d < n - c_1 \\log n$, the typical graph of diameter $d$ consists of an induced path of length $d$ and a highly connected block of order $n-d+3$. In the case $d > n - c_2 \\log n$, the typical graph has a completely different snake-like structure. We also extend the results to random graphs of diameter $d$ with edge probability $p$.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-11-21T22:18:12Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 KIM_YOUNJIN.pdf: 532935 bytes, checksum: d07b7137675a9bcc8a324b0553765ba3 (MD5)","Made available in DSpace on 2012-02-01T00:46:19Z (GMT). No. of bitstreams: 2 KIM_YOUNJIN.pdf: 531341 bytes, checksum: c67b8540f6a5ca81315b9097b17dc0ba (MD5) license.txt: 4059 bytes, checksum: dc42ed373ab2fd185eec9ce70491b3fa (MD5)","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2012-02-01T00:50:22Z Item is restricted until 2014-02-01T00:50:07Z","Item reinstated by Sarah Shreeves (sshreeve@illinois.edu) on 2014-02-01T11:00:26Z Item was in collections: Graduate Theses and Dissertations at Illinois (ID: 204) Dissertations and Theses - Mathematics (ID: 749) No. of bitstreams: 3 KIM_YOUNJIN.pdf: 531341 bytes, checksum: c67b8540f6a5ca81315b9097b17dc0ba (MD5) license.txt: 4059 bytes, checksum: dc42ed373ab2fd185eec9ce70491b3fa (MD5) KIM_YOUNJIN.pdf.txt: 122256 bytes, checksum: 0ff27d8c80eda8068a91b005e38f8f56 (MD5)","Item released from any restrictions by Sarah Shreeves (sshreeve@illinois.edu) on 2014-02-01T11:00:26Z"]},{"key":"dc:title","label":"Title","values":["Problems in extremal combinatorics"]}]}],"canonical_facts":{"dc:contributor":["Furedi, Zoltan","Kostochka, Alexandr V.","West, Douglas B.","Balogh, József"],"dc:creator":["Kim, Youn-Jin"],"dc:date":["2012-02-01T00:46:19Z","2014-02-01T11:00:26Z","2011-12"],"dc:description":["We consider a variety of problems in extremal graph and set theory. Given a property $\\Gamma$ and a family of sets ${\\mathcal F}$, let $f({\\mathcal F},\\Gamma)$ be the size of the largest subfamily of ${\\mathcal F}$ having property $\\Gamma$. Let $f(m,\\Gamma)$ be the minimum of $f({\\mathcal F},\\Gamma)$ over all families of size $m$ where $m$ is a positive integer. A family $\\mathcal{F}$ is {\\it $B_d$-free} if it has no subfamily $\\mathcal{F}'=\\{F_I: I \\subseteq [d]\\}$ of $2^d$ distinct sets such that for every $I,J \\subseteq [d]$, both $F_I \\cup F_J=F_{I \\cup J}$ and $F_I \\cap F_J = F_{I \\cap J}$ hold. A family $\\mathcal{F}$ is $a$-{\\it union-free} if $F_1\\cup \\dots \\cup F_a \\neq F_{a+1}$ whenever $F_1,\\dots,F_{a+1}$ are distinct sets in $\\mathcal{F}$. We prove a conjecture of Erd\\H os and Shelah that $f(m, B_2\\text{\\rm -free})=\\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\\text{\\rm -free})$ and $f(m,a\\text{\\rm -union-free})$. A graph $G$ is {\\it $F$-saturated } if it does not contain $F$ as a subgraph but the addition of any new edge creates at least one copy of $F$ in $G$. We focus on finding the minimum size of an $n$-vertex $F$-saturated graph, denoted by $\\sat(n,F)$. We prove $ \\sat(n,C_k) = n + \\frac{n}{k} + O((\\frac{n}{k^2}) + k^2)$ for all $n\\geq k\\geq 3$, where $C_k$ is a cycle with length $k$. We conjecture that our three constructions are optimal. We obtain the exact asymptotics for the number of $n$-vertex graphs of diameter $d$, extending earlier results to hold for almost all $d$ and $n$. Additionally, we find the typical structure of almost all $n$-vertex graphs with diameter of at least $d$. In the case $d < n - c_1 \\log n$, the typical graph of diameter $d$ consists of an induced path of length $d$ and a highly connected block of order $n-d+3$. In the case $d > n - c_2 \\log n$, the typical graph has a completely different snake-like structure. We also extend the results to random graphs of diameter $d$ with edge probability $p$.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-11-21T22:18:12Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 KIM_YOUNJIN.pdf: 532935 bytes, checksum: d07b7137675a9bcc8a324b0553765ba3 (MD5)","Made available in DSpace on 2012-02-01T00:46:19Z (GMT). No. of bitstreams: 2 KIM_YOUNJIN.pdf: 531341 bytes, checksum: c67b8540f6a5ca81315b9097b17dc0ba (MD5) license.txt: 4059 bytes, checksum: dc42ed373ab2fd185eec9ce70491b3fa (MD5)","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2012-02-01T00:50:22Z Item is restricted until 2014-02-01T00:50:07Z","Item reinstated by Sarah Shreeves (sshreeve@illinois.edu) on 2014-02-01T11:00:26Z Item was in collections: Graduate Theses and Dissertations at Illinois (ID: 204) Dissertations and Theses - Mathematics (ID: 749) No. of bitstreams: 3 KIM_YOUNJIN.pdf: 531341 bytes, checksum: c67b8540f6a5ca81315b9097b17dc0ba (MD5) license.txt: 4059 bytes, checksum: dc42ed373ab2fd185eec9ce70491b3fa (MD5) KIM_YOUNJIN.pdf.txt: 122256 bytes, checksum: 0ff27d8c80eda8068a91b005e38f8f56 (MD5)","Item released from any restrictions by Sarah Shreeves (sshreeve@illinois.edu) on 2014-02-01T11:00:26Z"],"dc:identifier":["http://hdl.handle.net/2142/29436"],"dc:language":["en"],"dc:rights":["Copyright 2011 Younjin Kim"],"dc:subject":["graphs","cycles","extremal graphs","minimal saturated graphs","diameter","random graphs","set families","boolean algebras"],"dc:title":["Problems in extremal combinatorics"],"dc:type":["Dissertation / Thesis","text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:27Z"}