In a proper vertex coloring of a graph a subgraph is colorful
if its vertices are colored with different colors. It is
well-known (see for example in Gyárfás (1980)) that in every
proper coloring of a k-chromatic graph there is a colorful
path Pk on k vertices. The first author proved in 1987 that
k-chromatic and triangle-free graphs have a path Pk which is
an induced subgraph. N.R. Aravind conjectured that these
results can be put together: in every proper coloring of a k-
chromatic triangle-free graph, there is an induced colorful
Pk. Here we prove the following weaker result providing some
evidence towards this conjecture: For a suitable function
f(k), in any proper coloring of an f(k)-chromatic graph of
girth at least five, there is an induced colorful path on k
vertices
