Lower bounds on the independence numbers of distance graphs with vertices in $\{-1, 0, 1\}^n$
A. R. Akhiiarov, A. V. Bobu, A. M. Raigorodskii
TL;DR
The paper advances the study of distance graphs with ternary vertices by establishing exponential lower bounds on the independence numbers $m(n,k_{-1},k_0,k_1,t)$ in the regime where $n$ grows large and the parameters scale linearly with $n$. It develops and combines multiple constructions—Alsvède–Hachatrjan (AH), Varshamov–Gilbert (VG), and glueing strategies (Glue, SuperGlue)—to derive a family of lower bounds, encapsulated in a sequence of theorems (VG, Glue, SuperGlue, Pairs) tied together by the auxiliary functions $h$ and $\mathrm{mdp}$. The results show that AH-like bounds dominate for small $t$, VG for large $t$, and Glue/SuperGlue in between, with Pairs offering optimality in certain odd-$t$ regimes. Numerical experiments illustrate the exponential rate gains and regime-specific dominance, highlighting the practical impact for asymptotic chromatic-number estimates of $\\\ ext{$\mathbb{R}^n$}$ under unit-distance constraints. Collectively, these bounds sharpen our understanding of independence numbers in high-dimensional distance graphs, informing both theoretical layering and potential applications in extremal combinatorics.
Abstract
This work is devoted to lower bounds on independence numbers of distance graphs with vertices in $\{-1,0,1\}^n$. The asymptotic case is studied, yielding new results over a broad range of parameters. Numerical results are presented, highlighting nontrivial relationships between the obtained bounds. Known upper bounds and their potential suboptimality are discussed separately.