bound sizes of integer-distance sets

This study investigates structural constraints on point sets in high-dimensional Euclidean and hyperbolic spaces where all pairwise distances are integers. By analyzing complete bipartite subgraphs in sphere-of-influence graphs and integrating techniques from discrete geometry, graph theory, and hyperbolic geometry, the work extends the classical Erdős–Anning theorem to high-dimensional hyperbolic spaces for the first time. It establishes a quantitative upper bound of $O(D(d+1)^D)$ on the cardinality of such point sets in Euclidean spaces of dimension three or higher. Key contributions include a finiteness criterion for integer-distance point sets, a proof that any such set containing $D+1$ points in general position has size bounded by its diameter and ambient dimension, and the demonstration that in multilateration, $D+1$ non-coplanar beacons suffice to restrict a target’s location to at most two possible positions.

This paper addresses the combinatorial characterization of subsets in finite Hamming spaces. We introduce a purely combinatorial notion—“subset combinatorial rank”—defined as the number of non-constant columns in the subset’s vector matrix, and establish tight upper and lower bounds on the sum of pairwise Hamming distances in terms of this rank. We further propose the new concept of “metric density” to characterize isometric images of subsets achieving minimal combinatorial rank. Our theoretical contributions are threefold: (i) the first algebraic–metric dual characterization of combinatorial rank; (ii) proof that uniformly column-distributed subsets—and all linear subspaces over any prime power field $mathbb{F}_q$—satisfy metric density; and (iii) demonstration that combinatorial rank is strictly constrained by metric density, with linear structures inherently attaining optimal rank compression. The work unifies extremal set theory, Hamming distance analysis, and finite-field vector space theory.

This work investigates combinatorial bounds for codes in general finite metric spaces, with a focus on conditions under which the classical Gilbert–Varshamov (GV) bound can be surpassed. By modeling codes as independent sets in proximity graphs, the authors develop a generalized GV framework applicable to arbitrary metric spaces and introduce novel concepts such as Ramsey–Sidorenko graphs and independence-forcing graphs. Their analysis demonstrates that local subgraph statistics alone are insufficient to exceed the GV bound; instead, global structural properties of the space are essential. Leveraging tools from graph theory, extremal combinatorics, entropy optimization via KKT conditions, and fractional packing techniques, they derive code bounds for both vertex-transitive and non-edge-transitive graphs, establishing density thresholds for several graph families. In particular, they prove that in Hamming spaces, no improvement over the GV bound is possible using only local information, and provide a tight upper bound based on fractional packing.

This paper addresses the efficient computation of the diameter of a point set in high-dimensional space. To overcome the practical inefficiency of traditional algorithms, we propose the first input-sensitive approximation algorithm for this problem. Our method integrates geometric pruning, iterative distance thresholding, and extremal direction sampling, coupled with exact verification and early termination mechanisms. It requires no preprocessing or parameter tuning, offering both theoretical guarantees and empirical robustness. Evaluated on extensive benchmark datasets, our algorithm significantly outperforms state-of-the-art methods in runtime while maintaining near-perfect (≈100%) accuracy. The key innovation lies in introducing input sensitivity to diameter computation—achieving ultra-fast performance on typical instances while retaining worst-case theoretical bounds. This balance makes it the empirically optimal diameter computation scheme at the time of publication.

This paper addresses the construction of random zero sets in $n$-point metric spaces satisfying a local growth condition, aiming to ensure that for any pair of points at distance $geq au$, one lies in the zero set while the other remains at bounded distance from it—achieving a constant lower bound on this separation probability. Methodologically, it refines the ARV rounding technique for zero-set construction under metric embeddings, establishing for the first time a quantitative link between local ball-volume ratios and separation probability. By integrating quasisymmetric embedding analysis with probabilistic metric geometry, the work derives that the maximum Euclidean distortion of any $n$-point set embeddable into $ell_1$ is $Theta(sqrt{log n})$. Consequently, it rigorously confirms that the integrality gap of the Goemans–Linial semidefinite program for the sparsest cut problem is also $Theta(sqrt{log n})$.

This work addresses the optimization of finite parameters in constructions related to counterexamples of the Erdős unit distance conjecture, aiming to improve the exponent in the lower bound on the number of unit distances determined by planar point sets. We formulate the parameter selection in Sawin’s explicit lower-bound proof as a nonlinear integer programming problem and introduce a lightweight, reproducible integer optimization framework. This framework integrates deterministic greedy strategies, a tailored integer evolutionary algorithm, and discrete recombination operators, enhanced by rational-number encoding and a repair mechanism. Implemented on standard hardware, our approach yields a new certificate δ = 0.0152628688…, establishing that for all sufficiently large n, u(n) > n^{1.0152}, thereby improving upon Sawin’s original result.

This work investigates the maximum asymptotic rate of the intersection of three Hamming balls of radius \( r = \alpha n \) whose centers are pairwise separated by distance \( k = \beta n \), in the limit as the code length \( n \to \infty \). Motivated by applications in list-recoverable coding and associative memory, the study combines tools from combinatorics, information theory, and asymptotic analysis to provide the first explicit characterization of this three-ball intersection rate as a function of the parameters \( \alpha \) and \( \beta \). In contrast to the well-understood two-ball case, the results uncover distinctive structural properties inherent to multi-ball intersections. Numerical experiments corroborate the theoretical predictions, demonstrating both their accuracy and the qualitative differences from the classical pairwise setting.

We study the computational complexity of exact cardinality-constrained minimum Riesz $s$-energy subset selection in finite metric spaces: given $n$ points, select $k<n$ points of minimum Riesz $s$-energy. The objective sums inverse-power pair interactions and therefore promotes well-separated subsets; as $s$ becomes large, it increasingly approaches a bottleneck criterion governed by the closest selected pair, linking it to minimum pairwise distance (MPD). Building on the general-metric NP-hardness result of Pereverdieva et al. (2025), we prove that NP-hardness persists for point sets in the Euclidean plane when $s$ is part of the input. In contrast, finite ultrametric spaces form an exact tractable regime: on rooted binary ultrametric trees with $n$ leaves, an optimal size-$k$ subset can be computed by dynamic programming in $O(nk^2)$ time. We also discuss the ordered one-dimensional Euclidean case, where the classical MPD objective admits simple dynamic programming, but the additive Riesz energy does not appear to allow the same state compression. Finally, we explain why one natural route to fixed-$s$ Euclidean hardness does not close: Fowler-style 3SAT gadgets, together with zeta-function bounds for far-field interactions, show why this approach still requires an exponent depending on $k$. Together, these results provide a compact complexity landscape for a natural diversity or dispersion objective, distinguishing Euclidean hardness, ultrametric tractability, and the ordered one-dimensional case.

This study addresses Erdős problem #272, which seeks to determine the maximum size $ t(N) $ of a family of subsets of $ \{1,\dots,N\} $ such that the intersection of any two distinct members is a nonempty arithmetic progression. By combining exhaustive computation—yielding exact values of $ t(N) $ for $ 3 \leq N \leq 12 $—with tools from structural graph theory and extremal set theory, the authors prove that Szabó’s construction is optimal among families sharing a common element (a kernel). They introduce a “deficiency-one” counting inequality and a structural theorem characterizing “bad pairs,” thereby establishing the first structural constraints supporting the kernel conjecture. This reduces the original problem to the existence of a kernel, significantly advancing the resolution of this classical question.