Some Consequences of Categorification

Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to $\mathbf{d}$-vectors, $\mathbf{g}$-vectors, and $F$-polynomials

A lower bound for the kk-multicolored sum-free problem in Zmn\mathbb{Z}^n_m

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$. This generalizes a result of Kleinberg-Sawin-Speyer for the case $k=3$ and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg-Sawin-Speyer. Both of these generalizations require several key new ideas