A group-theoretic approach to Shannon capacity of graphs and a limit theorem from lattice packings

We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003) proved that, in the limit $p\to \mathrm{\infty }$, the Shannon capacity of cycle graphs $\mathrm{\Theta }(C_p)$ converges to the fractional clique covering number, that is, ${\lim }_{p\to \mathrm{\infty }}p\mathrm{/}2-\mathrm{\Theta }(C_p)=0$. We strengthen this result by proving that the same is true for all fraction graphs: ${\lim }_{p\mathrm{/}q\to \mathrm{\infty }}p\mathrm{/}q-\mathrm{\Theta }(E_{p\mathrm{/}q})=0$. Here the fraction graph $E_{p\mathrm{/}q}$ is the graph with vertex set $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ in which two distinct vertices are adjacent if and only if their distance mod $p$ is strictly less than $q$. We obtain the limit via the group-theoretic approach. In particular, the independent sets we construct in powers of fraction graphs are subgroups (and, in fact, lattices). Our approach circumvents known barriers for structured (“linear”) constructions of independent sets of Calderbank-Frankl-Graham-Li-Shepp (1993) and Guruswami-Riazanov (2021).