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SUMMARY:Distinct degrees and homogeneous sets - Eoin Long (Birmingham)
DTSTART:20220602T133000Z
DTEND:20220602T143000Z
UID:TALK175313@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:In this talk I will discuss some recent work examining the ext
 remal relationship between two well-studied graph parameters: the order of
  the largest homogeneous set in a graph G and the maximal number of distin
 ct degrees appearing in an induced subgraph of G\, denoted respectively by
  hom (G) and f(G).\n\nOur main theorem improves estimates due to Bukh and 
 Sudakov and to Narayanan and Tomon and shows that if G is an n-vertex grap
 h with hom (G) at least n^{1/2} then f(G) > ( n / hom (G) )^{1 - o(1)}. Th
 e bound here is sharp up to the o(1)-term\, and asymptotically solves a co
 njecture of Narayanan and Tomon. In particular\, this implies that max { h
 om (G)\, f(G) } > n^{1/2 -o(1)} for any n-vertex graph G\, which is also s
 harp.\n\nThe relationship between these parameters is known to change when
  hom (G) < n^{1/2}. I hope to discuss the suspected relationship in this o
 ther region\, along with supporting results.\n\nJoint work with Laurentiu 
 Ploscaru.\n
LOCATION:MR14
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