# Meeting Details

Title: Distance graphs with finite chromatic number Algebra and Number Theory Seminar Hiren Maharaj, Clemson University The distance graph $G(D)$ with distance set $D=\{d_1,d_2, \ldots \}$ has the set $\mathbb{Z}$ of integers as vertex set, with two vertices $i,j \in \mathbb{Z}$ adjacent iff $|i-j|\in D$. Ruzsa, Tuza and Voigt proved that the chromatic number of $G(D)$ is finite whenever $\inf\{d_{i+1}/d_i\} >1$. In this talk I will discuss results obtained jointly with Jeong-Hyun Kang on distance graphs obtained by $p$-adic methods. For example, if for each prime $p$ we set $D(p)$ to be the set of all $p$-adic norms of elements of $D$ we show that the chromatic number is bounded above by $p^{\# D(p)}$. Thus this result complements that of Ruzsa, Tuza and Voigt.