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SUMMARY:A random Hall-Paige Conjecture  - Alexey Pokrovskiy (UCL)
DTSTART:20230525T133000Z
DTEND:20230525T143000Z
UID:TALK198319@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION: A rainbow subgraph in an edge-coloured graph is one in which 
 all\nedges have different colours. This talk will be about finding rainbow
 \nsubgraphs in colourings of graphs that come from groups. An old question
  of\nthis type was asked by Hall and Paige. Their question was equivalent 
 to the\nfollowing "Let G be a group of order n and consider an edge-colour
 ed Knn\,\nwhose parts are each a copy of G and with the edge {x\,y} colour
 ed by the\ngroup element xy. For which groups G\, does this coloured Knn c
 ontain a\nperfect rainbow matching?" This question is equivalent to asking
  "which\ngroups G contain a complete mapping'' and also "which multiplicat
 ion tables\nof groups contain transversals''.  Hall and Paige conjectured 
 that the\nanswer is "all groups in which the product of all the elements i
 s in the\ncommutator subgroup of G''. They proved that this is a necessary
  condition\,\nso the main part of the conjecture is to prove that that rai
 nbow matchings\nexist under their condition.\nThe Hall-Paige Conjecture wa
 s confirmed in 2009 by Wilcox\, Evans\, and Bray\nwith a proof using the c
 lassification of finite simple groups. Recently\,\nEberhard\, Manners\, an
 d Mrazovic found an alternative proof of the\nconjecture for sufficiently 
 large groups using ideas from analytic number\ntheory. Their proof gives a
  very precise estimate on the number of complete\nmappings that each group
  has.\nIn this talk\, a third proof of the conjecture will be presented us
 ing a\ndifferent set of techniques\, this time coming from probabilistic\n
 combinatorics. This proof only works for sufficiently large groups\, but\n
 generalizes the conjecture in a  new direction. Specifically we not only\n
 characterize when the edge coloured Knn contains a perfect rainbow\nmatchi
 ng\, but also when random subgraphs of it contain a perfect rainbow\nmatch
 ing. This extension has a number of applications\, such as to problems\nof
  Snevily\, Cichacz\, Tannenbaum\, Evans.\nThis is joint work with Alp Muye
 sser.\n
LOCATION:MR12
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