Papers

In this paper, by using $q$-identities and differential operator techniques, we establish two $q$-supercongruences modulo the square of a cyclotomic polynomial, which are associated with truncated double basic hypergeometric $q$-series.

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}\subseteq {V\brack k}$ and $\mathcal{G}\subseteq {V\brack \ell}$ are said to be cross $t$-intersecting if $\dim(F\cap G)\geq t$ for all $F\in\mathcal{F}$ and $G\in \mathcal{G}$. In this paper, we determine the extremal structures when $|\mathcal{F}||\mathcal{G}|$ attains the maximum value under the conditions $\dim\left(\cap_{F\in \mathcal{F}}F\right)<t$ and $\dim\left(\cap_{G\in \mathcal{G}}G\right)<t$.

It is well known that the number of partitions of $[n]$ without singletons equals the number of partitions of $[n]$ in which no block contains two cyclically adjacent elements $i,i+1\pmod{n}$. Bernhart remarked that there might be no simple bijection between these two classes. Although Callan later constructed an algorithmic bijection proving the stronger equidistribution of singletons and adjacencies, his construction proceeds through multiple rounds of exchanges. Therefore, Bernhart's remark may still retain some validity, as suggested by Chen and Wang. In this article, we address this remark by giving a direct “one-round” bijection between the two classes. Unlike Callan's bijection, our map is closely compatible with the refinement order on partitions: in one direction it only decomposes blocks, while its inverse only merges blocks, with a single exceptional pair when $n>2$ is even. We further observe that this exception is unavoidable, establishing the optimality of the bijection with respect to the refinement order. The specific local form of these operations — splitting off only singleton blocks and merging a singleton only with the block containing its cyclic neighbor — also ensures that the construction restricts, without modification, to a bijection between the corresponding classes of noncrossing partitions.

A graph $G$ is $F$-saturated if $G$ contains no copy of $F$ but $G+e$ contains a copy of $F$ for every missing edge $e$ of $G$. The saturation number $\sat(n,F)$ is the minimum number of edges in an $n$-vertex $F$-saturated graph. Motivated by a problem posed by Faudree, Ferrara, Gould, and Jacobson concerning $K_p\cup K_q\cup K_{q+1}$, we determine the saturation number and the unique extremal graph for $K_p\cup K_q\cup K_r$ whenever $2\le p\le q<r<p+q$ and $n$ is sufficiently large. Together with the previously known results for $r\ge p+q$ and for $r=q$, this completes the determination of the saturation number and the extremal graphs for unions of three cliques, for all sufficiently large $n$.

We study when statistically learnable latent structure can also be recovered efficiently, and how membership queries change the answer. An unknown support $A\subseteq\mathbb F_2^n$ has small additive doubling and is observed through a fixed set $B$ satisfying $|A\triangle B|\le\eta|A|$. We seek one linear subspace $V$ such that every compatible support $A$ is covered by few $V$-cosets and satisfies $|V|\le|A|$. For every $\eta<1$, polynomially many uniform samples suffice statistically, with cost polynomial in the doubling constant and proportional to $(1-\eta)^{-1}$; this radius dependence is sharp. Under a specified hardness assumption for learning parities with noise (search-LPN), however, no polynomial-time sample-only learner achieves even constant covering cost, including when the latent support is unique. At fixed structural parameters and the same constant covering budget, adding exact membership queries to $B$ permits polynomial-time recovery. The general query learner constructs a short structural list and uses fresh samples to select one common output through a majority-coverage rule. Persistent structured cores make this candidate construction possible. At doubling one, a complementary distinction appears at $\eta=1/3$: coarse recovery remains polynomial time, while exact recovery requires exponentially many accesses in the worst case when latent cardinality is unknown.

A Catalan multijection is a way to partition a set of $\binom{2n}{n}$ objects into equivalence classes of size $n+1$, where each equivalence class has exactly one distinct Catalan object. Isaak and Langley investigated when two multijections produced the same partitions in an unpublished manuscript. We propose some new multijections for which we get different partitions from previously discovered multijections.

In 2016, Straub proved that Euler's classic partition identity holds true for partitions with largest hook (perimeter) $n$. This inspired further study of the relationship between classical partitions and fixed perimeter partitions. We extend the study of parity bias inequalities, first introduced by Kim, Kim, and Lovejoy in 2020, to the fixed perimeter setting and show using combinatorial methods that fixed perimeter analogues of many classical parity bias results can be proven and generalized. We also extend these methods to prove similar inequalities for the fixed perimeter analogues of PED and POD partitions. We additionally develop recursive formulas for the number of perimeter $n$ partitions with odd parts distinct and even parts unrestricted and with even parts distinct and odd parts unrestricted.

Schubitopes are a family of generalized permutahedra arising as the Newton polytopes of both Schubert polynomials and Demazure characters of $\mathrm{GL}_n$. Monical, Tokcan, and Yong conjectured that Schubitopes are Ehrhart positive. We exhibit a counterexample to this conjecture, a Schubitope whose Ehrhart polynomial contains a negative coefficient in its monomial expansion. This example was found using GPT-5.6 Sol Pro in ChatGPT.

We give an explicit description, in terms of the quantum Bruhat graph, of the (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants $\langle \mathcal{O}(- \lambda), \mathcal{O}^{w}, \mathcal{O}_{u} \rangle_{d}$ for the (full) flag manifold $X = G/B$, where $\mathcal{O}(- \lambda)$ denotes the class in the (torus-equivariant) $K$-theory ring $K_{T}(X)$ of $X$ of the line bundle $\mathcal{O}_{X}(- \lambda) = G \times_{B} \mathbb{C}_{\lambda}$ over $X = G/B$ associated to a weight $\lambda \in W \varpi_i$ lying in the Weyl group orbit of a minuscule fundamental weight $\varpi_i$, and $\mathcal{O}_{u}$, $\mathcal{O}^{w}$ are the Schubert and opposite Schubert classes in $K_{T}(X)$ for $u, w \in W$. This result can be thought of as a partial generalization of the quantum $K$-theoretic divisor axiom, which we obtained in our previous work; our proof utilizes a generalization of the Chevalley formula in the (torus-equivariant) quantum $K$-theory ring $QK_{T}(X)$ of $X$, which computes the quantum product with the line bundle class $\mathcal{O}(- \lambda)$ associated to the weight $\lambda$ above.

The booksize $\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollobás and Erdős, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502–520] asked whether the same conclusion holds under the spectral condition $\lambda(G)>\lambda(T_{n,2})$, where $\lambda(G)$ is the spectral radius of the adjacency matrix. We answer this question in a strong form: every $n$-vertex graph $G\neq T_{n,2}$ with $\lambda(G)\ge\lambda(T_{n,2})$ satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\frac13\lambda(G),\,\lambda(G)-\frac n3,\,2\lambda(G)-n\Big\}. \] Consequently, the condition $\lambda(G)>\lambda(T_{n,2})$ forces $\mathrm{bk}(G)\ge\lfloor n/6\rfloor+1$. The middle term is a spectral improvement of Edwards' bound $\mathrm{bk}(G)\ge\frac{2m}{n}-\frac n3$, and all three bounds are best possible. These results come from the edge-spectral setting: every graph $G$ with $m$ edges and $\lambda(G)\ge\sqrt m$ that is not a complete bipartite graph satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\lambda(G)-\frac{2m}{3\lambda(G)},\,2\lambda(G)-\frac{2m}{\lambda(G)}\Big\}, \] which strengthens the bound $\mathrm{bk}(G)\ge\frac13\lambda(G)$ of Zhao, You, Zeng and Zhang. As an application of our method, we prove a triangle counting bound $t(G)\ge\frac13(\lambda(G)+1)(\lambda(G)^2-m)$, which improves the result of Bollobás and Nikiforov [J. Combin. Theory Ser B. (2007)]. Finally, we prove stability results at both thresholds: if $\lambda(G)\ge(\frac12-o(1))n$, then either $\mathrm{bk}(G)\ge(\frac16-o(1))n$ or $G$ can be made into $T_{n,2}$ by adding and deleting $o(n^2)$ edges; if $\lambda(G)\ge(1-o(1))\sqrt m$, then either $\mathrm{bk}(G)\ge(\frac13-o(1))\sqrt m$ or $G$ differs from a complete bipartite graph in $o(m)$ edges.

Designing high-performance tactical wireless networks under realistic operational constraints gives rise to challenging combinatorial optimization problems, where the evaluation of candidate solutions relies on detailed physical and traffic-aware models. Although classical metaheuristics such as Tabu Search offer effective mechanisms for exploring large search spaces, their computational cost remains high because numerous candidate moves must be evaluated at every iteration. In this paper, we propose a data-driven framework that improves the efficiency of Tabu Search by learning to guide its move selection process. Rather than altering the neighborhood structure, our approach exploits the information contained in the search trajectories generated during the optimization process. At each iteration, we record both improving and non-improving edge-based transformations together with a set of descriptive features capturing the structural, geometric, and performance characteristics of the network. This information is used to train a Graph Neural Network (GNN) that predicts the impact of candidate moves on the objective function. The trained model is then integrated into the Tabu Search algorithm to rank candidate transformations according to their predicted quality, thereby reducing the number of costly objective evaluations while maintaining an effective exploration of the search space. Experimental results on synthetic benchmark instances demonstrate that the proposed learning-assisted Tabu Search notably reduces computation time while consistently producing higher-quality solutions than the standard algorithm. These findings highlight the potential of combining machine learning with metaheuristics by leveraging the implicit knowledge embedded in search trajectories, paving the way for more efficient solution methods for large-scale network design problems.

We study a graph-based cellular automaton for HIV infection dynamics in lymph-node networks, originally introduced by Mukwembi. Each vertex represents a cell site that may be healthy, infected, or dead, and the evolution is controlled by a replacement parameter $R$, which determines whether a dead cell is replaced by an infected or a healthy cell according to the number of its infected neighbors. For a graph $G$, we introduce two extinction parameters. The parameter $\mathbf{hiv}(G)$ is the smallest value of $R$ for which extinction occurs for every admissible initial configuration, whereas $\mathbf{HIV}(G)$ is the smallest threshold such that extinction occurs for every replacement parameter greater than or equal to it. We prove the general bounds $2\leq \mathbf{hiv}(G)\leq \mathbf{HIV}(G)\leq \Delta(G)+1$ and characterize the extremal case $\mathbf{HIV}(G)=\Delta(G)+1$. We also show that the gap $\mathbf{HIV}(G)-\mathbf{hiv}(G)$ is unbounded and determine both parameters for some classical families of graphs. Finally, we study the dynamics of the model using the state-transition digraph of the system and the configurations whose trajectories converge to nontrivial periodic orbits. The results show that extinction depends not only on the replacement parameter but also on the structural properties of the underlying graph.

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $\alpha_3+\epsilon$ for every $\epsilon>0$, where $\alpha_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $\alpha_{\rm BOV}+\epsilon$ for every $\epsilon>0$, where $\alpha_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

It is well known that, given a Steiner triple system, a quasigroup can be formed by defining an operation $\cdot$ by the identities $x \cdot x = x$ and $x \cdot y = z$ where $z$ is the third point in the block containing the pair $\{x,y\}$. The same is true for a Mendelsohn triple system where the pair $(x,y)$ is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist and we call these Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $x \cdot (y \cdot x) = (x \cdot y) \cdot x$ but those associated with Latin directed triple systems need not. In a previous paper, [Discrete Mathematics 312 (2012), 597-607], we studied non-flexible Latin directed triple systems. In this paper we turn our attention to flexible Latin directed triple systems.

Han and Xiong recently extended the Gaussian binomial coefficient $\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}$ to positive rational $r$ and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at $r=1/2$. We prove a support-dominance theorem comparing rational parameters under an explicit divisibility condition. It settles the conjecture for every $r\geq 1/2$ and reduces the full conjecture to the unit fractions $r=\frac{1}{2m}$, only finitely many of which are nontrivial for each fixed $k$. A computer computation then verifies the conjecture for every positive rational $r$ and every $k\leq 200$. The theoretical results were autonomously produced and verified in Lean by AxiomProver.

A missing integer in a partition $\lambda$ is a positive integer less than the largest part of $\lambda$ that does not appear as a part in $\lambda$. In the recent paper On the number of missing integers in partitions, Bhoria, Eyyunni, and Santra examined the number of partitions (overpartitions) of $n$ with a fixed number of missing integers and established generating functions, identities, and congruences for them. In this article, we provide combinatorial proofs for several of their results.

A graph G with m edges has an antimagic orientation if its edges can be oriented and bijectively labelled 1, ..., m so that the oriented vertex sums, the labels on in-arcs minus the labels on out-arcs, are pairwise distinct. Hefetz, Mütze and Schwartz conjectured that every connected graph admits an antimagic orientation. We prove that this conjecture holds for every graph with a dominating clique.

Combinatorial discrepancy theory is a subject with roots in combinatorics, geometry, and number theory, and with numerous applications to mathematics and computer science. At its core, discrepancy theory is about dividing a collection of objects into two parts that are as balanced as possible. For example, given a collection of subsets of a finite universe, we may wish to color the elements with two colors so that each set is approximately evenly split. Other problems in discrepancy are more geometric in flavor, and ask for example, to assign signs to a collection of vectors, so that the sum of the signed vectors is as small as possible. Classical results, such as the Beck-Fiala theorem and Spencer's "six deviations" result, show that it is often possible to attain remarkably small discrepancy, often far smaller than what naive random colorings achieve. In recent years, discrepancy theory has undergone a transformation, driven by new algorithmic techniques and a rich interplay between probability, optimization, and convex geometry. These developments have led not only to new constructive proofs of foundational theorems, but also to several new results and research directions. This monograph aims to provide an accessible and unified introduction to these modern developments, with a focus on the core algorithmic and convex geometric ideas that have driven them. For several results, we provide new simpler analyses, while highlighting the intuition behind the proofs.

We prove bijectively a recursive excision formula for maps of arbitrary topology. This yields the topological recursion formulae governing their enumeration, already at the level of formal generating series and without any analyticity assumption. We extend the result to maps carrying self-avoiding loop models and to stuffed maps, i.e. a variant of maps allowing faces of arbitrary topology. Our results give a precise combinatorial meaning to all terms of the (blobbed) topological recursion known for (stuffed) maps, including the recursion kernel, which had so far been missing. The construction relies on the simple idea of iterating Tutte's algorithm until the topology changes. Equivalently, it can be interpreted as a pair-of-pants decomposition driven by a path issuing from the root of the first boundary, providing an analogue for maps of the Mirzakhani-McShane identity for hyperbolic surfaces.

A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68–90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.

We give new lower bounds for the Shannon capacities of small odd cycles: $\Theta(C_7)\geq3.258805369885\ldots$, $\Theta(C_{11})\geq5.294502522149\ldots$, $\Theta(C_{13})\geq6.302455083464\ldots$, $\Theta(C_{15})\geq7.301600534487\ldots$, $\Theta(C_{19})\geq9.357192705918\ldots$, $\Theta(C_{21})\geq10.342455853338\ldots$, and $\Theta(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

Let $G_n$ be the $n\times n$ square grid, and let $k\geq 2$. A set $D\subseteq V(G_n)$ is an \emph{exact-distance $k$-dominating set} if every vertex $v\in V(G_n)\setminus D$ has a vertex $u\in D$ with $d(u,v)=k$. We write $D_{\mathrm{opt}}^{(k)}(G_n)$ for the minimum cardinality of such a set. For every fixed $k$, consider the limit $ \delta_k= \lim_{n\to\infty} \frac{D_{\mathrm{opt}}^{(k)}(G_n)}{n^2}. $ We prove that, for every fixed \(k\geq 3\), $ \frac{1}{4k} \leq \delta_k \leq \frac{k-1}{3k^2-k-1}. $ For $k=2$, the exact value $\delta_2=1/9$ follows directly.

We determine the asymptotically best possible minimum vertex degree condition forcing a two-coloured $3$-graph to contain a colour-biased tight Hamilton cycle. This confirms a conjecture of Hàn, Lang, Marciano, Pavez-Signé, Sanhueza-Matamala, Treglown and Zárate-Guerén.

We introduce a new class of Riordan matrices corresponding to lattice walks with colored steps. We show that this class corresponds to a class of pseudo-involutions and then find both generating functions and explicit combinatorial formulas for entries of these Riordan matrices. We use these Riordan matrices to find an explicit combinatorial formula for the occurences of an arbitrary pattern in our lattice walks. Finally, we define a distributive lattice on our lattice walks and connect our new RNA matrices to Whitney numbers of lattices.

We study a restriction of the classical Erdős–Sós problem, the extremal number of trees, to the class of bipartite host graphs, both when only the order of the host is prescribed and when its two part-sizes are fixed. We give natural lower-bound constructions and formulate corresponding linear upper-bound conjectures. We apply a weighted variant of $k$-minimality to prove upper bounds for a broad family of trees including brooms, trees with part-sizes obeying certain inequalities, and all trees on at most seven vertices, resolving part of a problem of Caro, Patkós and Tuza up to additive constants. We also relate the fixed-part extremal number of a tree to the ordinary extremal number, and consider an oriented bipartite extremal function analogous to the Zarankiewicz function.

The Berge-Fulkerson Conjecture states that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. This conjecture has remained open since 1971, and several of its weakenings have been proposed over the years. Two of the most prominent are the Fan-Raspaud Conjecture and the Máčajová-Škoviera Conjecture. It is well known that the Berge-Fulkerson Conjecture implies the Fan-Raspaud Conjecture, which in turn implies the Máčajová-Škoviera Conjecture. These problems have been studied for years in the context of finite graphs, and many equivalences between them and other results have been established. However, little to nothing is known about them in the context of infinite graphs. In this paper, we investigate whether these conjectures remain valid in the infinite setting, establish their implications in analogy to the finite case, and prove that their finite versions are equivalent to their respective infinite versions.

We study polynomial zero-avoidance over finite vector spaces by means of slice rank and partition rank. We first make the support-entropy method effective by showing how a finite dual certificate yields an explicit entropy gap whenever the coefficient support admits no probability distribution with uniform marginals. For the quadratic elementary symmetric polynomial over fields of characteristic three, the ternary structure of the coefficient support gives a certificate with optimal normalized margin and a uniform analytic bound for the corresponding higher-degree Erdős–Ginzburg–Ziv constant, avoiding a separate optimization for each field. We then use partition rank to handle solutions in pairwise distinct variables. Equality profiles are encoded by contracted local tensors, reducing the global problem to finitely many slice-rank estimates. Applying this reduction on the multiplicative torus gives restricted-alphabet zero-sum bounds with exponential base below the alphabet size. Coordinatewise inversion and support stratification then yield, to the best of our knowledge, the first nontrivial exponential bound for the higher-degree Erdős–Ginzburg–Ziv problem over $\mathbb{F}_5^n$ associated with the fourth elementary symmetric polynomial.

Place the vertices of a finite graph at projective points. Each edge defines a span map to $\mathrm{Gr}(2,n)$; the pulled-back Chern classes generate a graded algebra $A_G^{(n)}$. This assignment is a covariant graph functor, so the abstract graded-algebra type is a graph invariant. In ambient dimension four, the Hilbert series is incomparable with the chromatic and Tutte polynomials. On five vertices in dimension three, the $34$ graph classes yield $33$ algebra types, strictly refining both classical polynomials. For every tree $T$, we obtain a closed Hilbert-series formula depending only on $|V(T)|$ and $n$, while $A_T^{(3)}$ determines $T$ up to isomorphism. More precisely, its cubic relation data is a complete tree invariant of polynomial size. Finally, the graphical configuration space embeds as a dense open in the picture variety, and picture spaces with equal additive homology can have nonisomorphic edge-Chern algebras.

Helly, Carathéodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carathéodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.