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UID:e452912d5a149911831136405155329a
CATEGORIES:Discrete Math
CREATED:20220928T134833
SUMMARY:Quantitative problems in infinite graph Ramsey theory
LOCATION:Hill Center Room 705
DESCRIPTION:Abstract: Two well-studied problems in Ramsey theory are (1) given a graph
G on n vertices, what is the smallest integer N such that there is a monoch
romatic copy of G in every 2-coloring of a complete graph on N vertices, an
d (2) given a directed acyclic graph D on n vertices, what is the smallest
integer N such that there is a copy of D in every tournament on N vertices.
Note that for both problems, the family of trees has turned out to be an
interesting special case, each with a long history and a relatively recent
resolution (for sufficiently large n).\nWe consider quantitative analogues
of these problems in the infinite setting; that is, (1) given a countably i
nfinite graph G what is the supremum of the set of real numbers r such that
in every 2-coloring of the complete graph on the natural numbers there is
a monochromatic copy of G whose vertex set has upper/lower density at least
r, and (2) given a countably infinite directed acyclic graph D what is the
supremum of the set of real numbers r such that in every tournament on the
natural numbers there is a copy of D whose vertex set has upper/lower dens
ity at least r? As it relates to these problems, I will discuss two very s
urprising results.\nBased on joint work with Alistair Benford, Jan Corsten,
and Paul McKenney.\n
X-ALT-DESC;FMTTYPE=text/html:**Abstract**: Two well-st
udied problems in Ramsey theory are (1) given a graph G on n vertices, what
is the smallest integer N such that there is a monochromatic copy of G in
every 2-coloring of a complete graph on N vertices, and (2) given a directe
d acyclic graph D on n vertices, what is the smallest integer N such that t
here is a copy of D in every tournament on N vertices. Note that for
both problems, the family of trees has turned out to be an interesting spec
ial case, each with a long history and a relatively recent resolution (for
sufficiently large n).

We consider
quantitative analogues of these problems in the infinite setting; that is,
(1) given a countably infinite graph G what is the supremum of the set of r
eal numbers r such that in every 2-coloring of the complete graph on the na
tural numbers there is a monochromatic copy of G whose vertex set has upper
/lower density at least r, and (2) given a countably infinite directed acyc
lic graph D what is the supremum of the set of real numbers r such that in
every tournament on the natural numbers there is a copy of D whose vertex s
et has upper/lower density at least r? As it relates to these problem
s, I will discuss two very surprising results.

Based on joint work wi
th Alistair Benford, Jan Corsten, and Paul McKenney.

CONTACT:Louis DeBiasio (Miami University)
DTSTAMP:20230925T041234Z
DTSTART;TZID=America/New_York:20221003T140000
DTEND;TZID=America/New_York:20221003T150000
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