Thursday, February 1, 2018
Time  Items 

All day 

4:00pm 
02/01/2018  4:00pm to 5:00pm For a graph G with m vertices, Mr(G,n) is the minimumnumber of monochromatic copies of G in any redgecoloring of Kn and is called \textitthe Ramsey multiplicity. A graph G with e edges is called \textitcommon if \displaystyle{\limn → ∞ \fracMr(G\;n){\binomnm(m!)/(Aut(G))} = 21e}, \textiti.e.\ the proportion of monochromatic copies of G in any coloring of a complete graph is asymptotically minimized by the random coloring. We introduce the \textitAntiRamsey Multiplicity rb(G,n,r), which is defined as the maximum number of \textitrainbow copies of agraph G in any redgecoloring of Kn. A graph is ranticommon if \displaystyle{\limn → ∞ r/b(G,n,r){\binomnm(m!)/(Aut(G))} = \frac{\binomree!}re}, \textiti.e. the proportion of rainbow copies of G is asymptotically maximized by the random coloring. A graph is anticommon if it is ranticommon for any r greater than or equal to the graph's size. We show that matchings, stars and disjoint stars are anticommon. We also prove that for any c such that c+(1c)\log(1c) ≥ (2)/(m1) + 1/12\binomm22, if e ≥ c\binomm2, then G is not \binomm2anticommon. Using blowup constructions, we show that C4 is not 4anticommon and C5 is not 5anticommon. Location: 