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Exact colorings of graphs
Conjecture For
, let
be the statement that given any exact
-coloring of the edges of a complete countably infinite graph (that is, a coloring with
colors all of which must be used at least once), there exists an exactly
-colored countably infinite complete subgraph. Then
is true if and only if
,
, or
.
![$ c \geq m \geq 1 $](/files/tex/6e9b0747762b7ed29d72bfb8ff83a9110f286cd5.png)
![$ P(c,m) $](/files/tex/14873868ae1a8235d738e7111ea73071b27343f9.png)
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ m $](/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png)
![$ P(c,m) $](/files/tex/14873868ae1a8235d738e7111ea73071b27343f9.png)
![$ m=1 $](/files/tex/6f40cff00ac40c80d97567981a52eb493fb21fc5.png)
![$ m=2 $](/files/tex/9e690d9c19c2d925c1ba684b20c01ac17320a0a5.png)
![$ c=m $](/files/tex/515f3cb02a33bd92f10c6539c812479e94a53a11.png)
Stacey and Weidl have shown that given , there is an integer
such that
is false for all
.
Bibliography
* M. Erickson, ``A Conjecture Concerning Ramsey's Theorem,'' Discrete Mathematics 126, 395--398 (1994); MR 95b:05209
A. Stacey and P. Weidl, "The Existence of Exactly m-Coloured Complete Subgraphs," J. of Combinatorial Theory, Series B 75, 1-18 (1999)
* indicates original appearance(s) of problem.