  The Axelrod model is a spatial stochastic model for the dynamics of cultures
which includes two important social factors: social influence, the tendency of
individuals to become more similar when they interact, and homophily, the
tendency of individuals to interact more frequently with individuals who are
more similar. Each vertex of the interaction network is characterized by its
culture, a vector of $F$ cultural features that can each assumes $q$ different
states. Pairs of neighbors interact at a rate proportional to the number of
cultural features they have in common, which results in the interacting pair
having one more cultural feature in common. In this article, we continue the
analysis of the Axelrod model initiated by the first author by proving that the
one-dimensional system fixates when $F\leq cq$ where the slope satisfies the
equation $e^{-c}=c$. In addition, we show that the two-feature model with at
least three states fixates. This last result is sharp since it is known from
previous works that the one-dimensional two-feature two-state Axelrod model
clusters.
