A coloring of the constituents of a combinatorial structure is k-finite
(k-bounded) if it uses only k colors (uses each color
only k times). A combinatorial structure is
homogeneously (heterogeneously) colored if every
constituent is colored the same (differently). A
Ramsey-type theorem asserts that every k-finite
coloring of a certain structure induces a homogeneous
substructure of a certain size. An anti-Ramsey type
theorem asserts that that every k-bounded coloring of a
certain structure induces a heterogeneous substructure
of a certain size. Clearly k-finiteness or something
like it is necessary for the Ramsey theorems, otherwise
the structure may be colored heterogeneously.
Similarly k-boundedness is necessary for the
anti-Ramsey type theorems.
We'll talk a little about anti-Ramsey theorems in general and thresholds
for anti-Ramsey properties in random graphs.