  In Stanley's seminal 1995 paper on the chromatic symmetric function, he
stated that there was no known graph that was not contractible to the claw and
whose chromatic symmetric function was not $e$-positive, namely, not a positive
linear combination of elementary symmetric functions. We resolve this by giving
infinite families of graphs that are not contractible to the claw and whose
chromatic symmetric functions are not $e$-positive. Moreover, one such family
is additionally claw-free, thus establishing that the $e$-positivity of
chromatic symmetric functions is in general not dependent on the existence of
an induced claw or of a contraction to a claw.
