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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:<p style="margin: 9pt 0in 0.0001pt;"><strong>Abstract</strong>: 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.&nbsp; 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).</p><p style="margin: 9pt 0in 0.0001pt;">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?&nbsp; As it relates to these problem
 s, I will discuss two very surprising results.</p><p>Based on joint work wi
 th Alistair Benford, Jan Corsten, and Paul McKenney.</p>
CONTACT:Louis DeBiasio (Miami University)
DTSTAMP:20260828T162837
DTSTART;TZID=America/New_York:20221003T140000
DTEND;TZID=America/New_York:20221003T150000
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