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SUMMARY:Triangle factors in random graphs  - Oliver Riordan (Oxford)
DTSTART:20190225T160000Z
DTEND:20190225T170000Z
UID:TALK119128@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:The Erdös—Rényi or `binomial’ random graph G(n\,p) consi
 sts of n vertices\, with each pair connected by an edge with probability p
 \, independently of the others. The nature of the model means that `local
 ’ properties (such as individual vertex degrees) tend to be relatively e
 asy to study\, whereas `global’ properties (such as the size of the larg
 est component) are much harder. An interesting class of questions relates 
 one to the other. For example\, if p=p(n) is chosen so that G(n\,p) has wh
 p (`with high probability’\, i.e.\, with probability tending to 1 as n t
 ends to infinity) minimum degree at least 1\, does it also have (whp) the 
 global property of connectedness? The answer is yes\, as shown already by 
 Erdös and Rényi in 1960. What about minimum degree 2 and containing a Ha
 milton cycle? Again yes\, as shown by Komlós and Szemerédi in 1983. What
  about every vertex being in a triangle\, and the graph containing a trian
 gle factor\, i.e.\, a set of n/3 disjoint triangles covering all the verti
 ces? This question turned out to be much harder\, and was eventually answe
 red (approximately) by Johansson\, Kahn and Vu in 2008.\n\nIn this talk I 
 will describe at least some aspects of the proof of the last result\, as w
 ell as a related recent development. The aim is not so much to present par
 ticular results\, but rather to give a flavour of the range of methods tha
 t are used in studying this type of problem.\n\nThe colloquium is followed
  by a wine reception in Central Core.\n\n\n\n
LOCATION:MR3\, CMS
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