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UID:0c4440e3082add738904f8a1ddc0e38e
CATEGORIES:Discrete Math
CREATED:20230223T091755
SUMMARY:On the evolution of triangle-free graphs in the ordered regime 
LOCATION:Hill Center Room 705
DESCRIPTION:Abstract: Erd?s-Kleitman-Rothschild proved that the number of triangle-free
  graphs on n vertices is asymptotic to the number of bipartite graphs; or i
 n other words, a typical triangle-free graph is a random subgraph of a near
 ly balanced complete bipartite graph.  Osthus-Promel-Taraz extended this re
 sult to much lower densities: when m >(\sqrt{3}/4 +eps) n^{3/2} \sqrt{\l
 og n}, a typical triangle-free graph with m edges is a random subgraph of s
 ize m from a nearly balanced complete bipartite graph (and this no longer h
 olds below this threshold). \nWhat do typical triangle-free graphs at spars
 er densities look like and how many of them are there? We consider what we 
 call the "ordered" regime, in which typical triangle-free graphs are not bi
 partite but do align closely with a nearly balanced bipartition.  In this r
 egime we prove asymptotic formulas for the number of triangle-free graphs a
 nd give a precise probabilistic description of their structure.  This leads
  to further results such as determining the threshold at which typical tria
 ngle-free graphs are q-colorable for q >=3, determining the threshold fo
 r the emergence of a giant component in the complement of a max-cut, and ma
 ny others.  \nThis is joint work with Matthew Jenssen and Aditya Potukuchi.
 \n
X-ALT-DESC;FMTTYPE=text/html:<p style="margin: 0in;"><strong>Abstract</strong>: Erd?s-Kleitman-Rothschil
 d proved that the number of triangle-free graphs on n vertices is asymptoti
 c to the number of bipartite graphs; or in other words, a typical triangle-
 free graph is a random subgraph of a nearly balanced complete bipartite gra
 ph.&nbsp; Osthus-Promel-Taraz extended this result to much lower densities:
  when m &gt;(\sqrt{3}/4 +eps) n^{3/2} \sqrt{\log n}, a typical triangle-fre
 e graph with m edges is a random subgraph of size m from a nearly balanced 
 complete bipartite graph (and this no longer holds below this threshold).&n
 bsp;</p><p style="margin: 0in;">What do typical triangle-free graphs at spa
 rser densities look like and how many of them are there? We consider what w
 e call the "ordered" regime, in which typical triangle-free graphs are not 
 bipartite but do align closely with a nearly balanced bipartition.&nbsp; In
  this regime we prove asymptotic formulas for the number of triangle-free g
 raphs and give a precise probabilistic description of their structure.&nbsp
 ; This leads to further results such as determining the threshold at which 
 typical triangle-free graphs are q-colorable for q &gt;=3, determining the 
 threshold for the emergence of a giant component in the complement of a max
 -cut, and many others.&nbsp;&nbsp;</p><p style="margin: 0in;">This is joint
  work with Matthew Jenssen and Aditya Potukuchi.</p>
CONTACT:Will Perkins (Georgia Tech)
DTSTAMP:20260926T121027
DTSTART;TZID=America/New_York:20230227T140000
DTEND;TZID=America/New_York:20230227T150000
SEQUENCE:0
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