arXiv subjects

Guillaume Lecomte

Publications and source records attributed to Guillaume Lecomte.

4 recordsLinked to original sources

arXiv2026

Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity

We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let $\omega(G)$ denote the largest size of a pairwise noncommuting subset of a group $G$, let $a(G)$ be the least size of an abelian cover, and define $h(n)=\sup\{a(G):\omega(G)\le n\}$. We prove the quantitative estimate $\log_2 h(n)=n/2+O(\sqrt{n}\,(\log(n+2))^3)$, and hence $h(n)^{1/n}\to\sqrt{2}$. Extraspecial $2$-groups give the matching lower bound. For the upper bound, we reduce to finite groups by isoclinism, analyze finite $p$-groups through a central series of the derived subgroup and alternating commutator forms, control interactions between central factors, and then pass through Sylow decomposition and the Fitting subgroup at polynomial cost. The argument also determines the same sharp exponential rate for the least possible index of an abelian subgroup and shows that asymptotic extremality is concentrated in $2$-groups. This result resolves Erd\H{o}s Problem #117 at the level of its sharp exponential asymptotics.

math.GR↗

arXiv2026

Connected Counterexamples for Target Ramsey Numbers

Chartrand and Zhang asked whether there exists a graph $G$ without isolated vertices whose target Ramsey number satisfies $\mathrm{TR}(G) > \mathrm{R}(G)$. We answer the question affirmatively, even under strong structural restrictions. If $\Gamma_t = P_t \square P_t$ is the square $t \times t$ grid, then the elementary counting bound of Chartrand and Zhang gives $\mathrm{TR}(\Gamma_t) \geq 2t(t-1)+1$, whereas a theorem of Mota, Sarkozy, Schacht and Taraz gives $\mathrm{R}(\Gamma_t) = (3/2 + o(1))t^2$. Consequently, $\mathrm{TR}(\Gamma_t) > \mathrm{R}(\Gamma_t)$ for all sufficiently large $t$, so there are infinitely many connected, planar, bipartite counterexamples of maximum degree four. Higher-dimensional grids show that the ratio $\mathrm{TR}(G)/\mathrm{R}(G)$ is unbounded on connected bipartite graphs, while each fixed-dimensional witnessing family has bounded maximum degree. We also record an independent construction of disconnected counterexamples from the theorem of Burr, Erdos and Spencer on Ramsey numbers of multiple copies: if a fixed graph $H$ satisfies $e(H) > 2v(H) - \alpha(H)$, then $\mathrm{TR}(qH) > \mathrm{R}(qH)$ for every sufficiently large $q$.

math.CO↗

arXiv2026

The Certification Limits of KS-Type Layer Relaxations: A Square-Root Ceiling and Its Breakdown

A separable bilinear layer relaxation assigns a population cost to each layer and a nonpositive bilinear coupling to each pair of layers, then excludes a profile span whenever the relaxed minimum exceeds a reference value. We ask what a Kuznetsov-Sahinidis-type relaxation can certify at best. Let gamma = g(1) be the singleton-layer cost. For every valid relaxation with gamma > -1/2, every reference bounded per particle, and every underlying system satisfying two elementary low-energy witness conditions, we prove that the certified deficit N - rho(N) is at most of order sqrt(N), with an explicit bound depending only on the reference bound and gamma. Thus no relaxation in this class can certify a larger asymptotic deficit, independently of the detailed layer cost, interaction kernel, or reference. The threshold is sharp in the abstract witness class: at gamma = -1/2 we construct a valid relaxation with a bounded sound reference and deficit N - 1. The geometric Lennard-Jones system behaves differently. A three-atom chain restores the square-root ceiling at the endpoint, while sufficiently negative singleton costs allow valid relaxations with linear deficit. For still lower costs, beyond the stability constant, no sound relaxation excludes any span. These regimes define a certification spectrum for layer relaxations and reveal a reentrant transition between no exclusion, linear exclusion, and the universal square-root ceiling.

math.OC↗

arXiv2026

A certified refinement and asymptotic analysis of the Kuznetsov-Sahinidis diameter bound for Lennard-Jones clusters

Kuznetsov and Sahinidis (J. Glob. Optim., 2025) prove distance bounds that confine optimal Lennard-Jones clusters and shrink the search region of deterministic solvers; their diameter bound charges each unit-width layer the loosest internal energy $-\binom{n}{2}$. We replace this by a certified estimate built from their own subset inequality and the proven minima $V_5^*$, $V_6^*$, and minimize the resulting layer bound over population profiles. Centred decreasing profiles supply candidate minima; an arrangement-free relaxation, whose only classical ingredient is the rearrangement inequality for sequences, closes every certificate over all profiles; and all comparisons are re-verified in directed-rounding arithmetic. For $5 \le N \le 200$ this certifies a one-layer improvement of the published bound at 92 sizes. The improvement resolves no open global-optimization case: it is a rigorous tightening of a published a priori bound, with a precise account of the mechanism. A direct downstream test on the only tractable sizes ($N \le 6$) finds the diameter box is not the binding resource for a deterministic solver there, and no solver runs at the sizes the refinement affects ($N \ge 38$); we therefore present the result as a theoretical note. We also derive the asymptotic form of the bound, $\rho_{KS} = N - \Theta(\sqrt{N})$, resolving a point left open by Kuznetsov and Sahinidis; the gain of the refinement itself grows like $\Theta(\sqrt{N})$ layers, with an exact asymptotic constant.

math.OC↗