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CATEGORIES:Combinatorics and Probability Seminar
SUMMARY:Tight Ramsey bounds for multiple copies of a graph
- Matija Bucic\, Princeton
DTSTART:20220113T150000Z
DTEND:20220113T160000Z
UID:TALK4735AT
URL:/talk/index/4735
DESCRIPTION:The Ramsey number r(G) of a graph G is the smalles
t integer n such that any\n2-colouring of the edge
s of a clique on n vertices contains a monochromat
ic\ncopy of G. Determining the Ramsey number of G
is a central problem of Ramsey theory with a long
history. Despite this there are very few classes o
f\ngraphs G for which the value of r(G) is known e
xactly. One such family\nconsists of large vertex
disjoint unions of a fixed graph H\, we denote suc
h\na graph\, consisting of n copies of H by nH. Th
is classical result was proved\nby Burr\, Erdős an
d Spencer in 1975\, who showed r(nH)=(2|H|−α(H))n+
c\, for\nsome c=c(H)\, provided n is large enough.
Since it did not follow from their\narguments\, B
urr\, Erdős and Spencer further asked to determine
the number of\ncopies we need to take in order to
see this long term behaviour and the\nvalue of c.
More than 30 years ago Burr gave a way of determi
ning c(H)\,\nwhich only applies when the number of
copies n is triple exponential in |H|.\nWe obtain
an essentially tight answer to this very old prob
lem of Burr\,\nErdős and Spencer by showing that t
he long term behaviour occurs already\nwhen the nu
mber of copies is single exponential.\n\nJoint wor
k with B. Sudakov.
LOCATION:LTA\, additional zoom link: https://bham-ac-uk.zoo
m.us/j/87124623071
CONTACT:Johannes Carmesin
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