We study a long standing conjecture on the necessary and sufficient
conditions for the compatibility of multi-state characters: There exists a
function f(r) such that, for any set C of r-state characters, C is
compatible if and only if every subset of f(r) characters of C is
compatible. We show that for every r≥2, there exists an incompatible set
C of ⌊2r⌋⋅⌈2r⌉+1r-state
characters such that every proper subset of C is compatible. Thus, f(r)≥⌊2r⌋⋅⌈2r⌉+1 for every r≥2.
This improves the previous lower bound of f(r)≥r given by Meacham (1983),
and generalizes the construction showing that f(4)≥5 given by Habib and
To (2011). We prove our result via a result on quartet compatibility that may
be of independent interest: For every integer n≥4, there exists an
incompatible set Q of
⌊2n−2⌋⋅⌈2n−2⌉+1 quartets over
n labels such that every proper subset of Q is compatible. We contrast this
with a result on the compatibility of triplets: For every n≥3, if R is
an incompatible set of more than n−1 triplets over n labels, then some
proper subset of R is incompatible. We show this upper bound is tight by
exhibiting, for every n≥3, a set of n−1 triplets over n taxa such
that R is incompatible, but every proper subset of R is compatible