Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.
The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient $\ell_1$ metric), is isometric to $\mathbb{Z}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space -- the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes -- and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of $\mathbb{Z}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)\Delta_n$ in $\mathbb{R}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.
Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erd\H{o}s conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by S\'ark\"ozy and later again by Ruzsa, who found an elegant construction showing that $D(N)\geq c\cdot N^{0.733077\dots}$, with an absolute constant $c>0$. His approach was subsequently refined, leading to the previously best known lower bound with exponent $0.7334117\dots$ due to Beigel-Gasarch and, independently, Lewko. However, in the original paper Ruzsa observed that $3/4$ seems to be the natural barrier of his approach. In this paper we develop a new construction leading to the lower bound \[ \liminf_{N\to\infty}\frac{\log D(N)}{\log N} \geq \alpha_*:= 0.7527964558\ldots; \] thus crossing the natural exponent-$3/4$ barrier of Ruzsa's method. The value $0.7527964558\ldots$ arises from a simple optimisation problem and appears to be the limit of the new approach.
We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\varepsilon>0$, there exists a set of primes $q$ of positive relative density for which $\mathbb F_q^2$ contains an induced point-line matching of size $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstra\"ete and Zhang.
Fix integers $0\leq k<n$. Let $F_{n,k}(N)$ be the least size of a finite set $B\subset\Z^n$ that contains a filled axis-parallel cube $k$-skeleton centered at each point of some $N$-point set. We prove that $F_{n,k}(N)$ has order $N^{1-(n-k)/(2n^2)}$, with constants depending only on $n$ and $k$. Thornton proved the upper bound and lower bounds with every smaller exponent; the endpoint lower bound was open for $k\geq1$. For square boundaries in $\Z^2$, the exponent is $7/8$. A midpoint count and Shearer's inequality handle large radii; induction in lattice cells handles small radii.
Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $\delta_n\le\pi/4$. In the opposite direction, ChatGPT combines the passage from small Gromov--Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ \delta_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.
We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudl\'ak. In particular, $R(33, t) \geq t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s, t) \geq t^c$ for some fixed $s$ and some $c>2$. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl--Wilson bound from $1/4$ to $1$, while being almost as simple to describe.