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Danila Cherkashin

Publications and source records attributed to Danila Cherkashin.

At least 19 recordsLinked to original sources
arXiv2026

Envelopes of upper bounds for nonbinary constant-weight and constant-composition codes

Deriving upper bounds on code size from existing bounds is a classical approach in coding theory, dating back to the seminal results of Elias, Bassalygo, and Levenshtein. We study a framework encompassing the Bassalygo--Elias and Levenshtein inequalities for binary and nonbinary (constant-weight) codes and provides certain generalizations. The asymptotic cost of transferring a bound between different symbol compositions is expressed in terms of mutual information, yielding an information-theoretic optimal transport formulation. We determine the optimal permutation-transport cost between arbitrary compositions in terms of their least common majorant in the majorization order. Specializing to symmetric constant-weight compositions yields explicit transport profiles. As a byproduct, we establish unimodality of the asymptotic constant-weight rate as a function of the relative weight. We prove that the resulting closure operators are idempotent and that applying transport before outer Bassalygo--Elias averaging leaves the unrestricted bound obtained from the same input unchanged. We also establish necessary and sufficient conditions for an upper bound to be a fixed point of the closure operator. Since the asymptotic rate function is an upper bound for itself and it is a fixed point, we conclude the Schur concavity of the constant-composition rate function. Finally, we survey existing upper bounds for binary and nonbinary constant-weight and constant-composition codes, combine them into optimized envelopes within the transport framework, and obtain improved theoretical and numerical bounds.

cs.IT↗
arXiv2026

A short proof of Oblakov's theorem

We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic graph has the same numbers of balanced and unbalanced vertices.

math.CO↗
arXiv2026

On uniform eventowns

Suppose that $n=2m$, $k=2t$ and $n > 10 k^7$. We show that if a family $\mathcal F$ of $k$-subsets of an $n$-set has only even pairwise intersections then $|\mathcal F| \leq \binom{m}{t}$. Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for $n > n_{FT}(k)$, where $n_{FT}(k)$ is at least exponential. The main technique is Delsarte linear programming.

math.CO↗
arXiv2026

Short proofs of three combinatorial results in the Johnson scheme

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims that if $n > n_0(k)$ and there are an $(n,k,L)$-system and an $(n,k,\{0,\dots,k-1\}\setminus L)$-system whose sizes have product $\binom{n}{k}$, then they are a $t$-intersecting family and a Steiner system $S(t,k,n)$ for some $t$.

math.CO↗
arXiv2026

Universal Ahlfors--David regularity of Steiner trees

The celebrated Steiner tree problem is the problem of finding a set $St$ of minimum one-dimensional Hausdorff measure $H$ (length) such that $St \cup \mathcal{A}$ is connected, where $\mathcal{A} \subset \mathbb{R}^d$ is a given compact set. Paolini and Stepanov provided very general existence and regularity results for the Steiner problem. Their main regularity result is that under a natural assumption, $H(St) < \infty$, for almost every $\varepsilon>0$ the set $St_\varepsilon := St\setminus B_\varepsilon(\mathcal A)$ is an embedded finite forest (acyclic graph). We give a quantitative regularity result by proving that the set $St_\varepsilon$ is Ahlfors--David regular with constants that depend only on $d$ (and not on $\mathcal{A}$). Namely, for $d > 2$, every $\varepsilon > 0$, every $x \in St_\varepsilon$, and every choice of $ρ\in (0,1)$, we have \[ \frac{H \left (St_\varepsilon \cap B_{ρ\varepsilon}(x) \right) }{\varepsilon} \leq \left ( \frac{144d}{1-ρ} \right) ^{d-2}. \] As a corollary, we obtain a density-type result, i.e. that the set $St_\varepsilon \cap B_{ρ\varepsilon}(x)$ consists of at most \[ \left ( \frac{144d}{1-ρ} \right) ^{d-1} \] line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.

math.MG↗
arXiv2025

On the chromatic numbers of Johnson type graphs

A Johnson type graph $J_{\pm}(n,k,t)$ is a graph whose vertex set consists of vectors from $\{-1,0,1\}^n$ of the length $\sqrt{k}$ and edges connect vertices with scalar product $t$. The paper determines the order of growth of the chromatic numbers of graphs $J_\pm(n,2,-1)$ and $J_\pm(n,3,-1)$ (logarithmic on $n$), and also $J_\pm(n,3,-2)$ (double logarithmic on $n$).

math.CO↗
arXiv2025

On branching points in the Gilbert-Steiner problem

The Gilbert--Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use additional (branching) point in a transport plan. A specific feature of the problem is that the cost of transporting a mass $m$ along a segment of length $l$ is equal to $l \times m^p$ for a fixed $0 < p < 1$ and segments may end at points not belonging to the supports of given measures (branching points). Main result of this paper determines all pairs of $(p,d)$ for which the Gilbert--Steiner problem in $\mathbb{R}^d$ admits only branching points of degree 3. Namely, it happens if and only if $d = 2$ or $p < 1/2$.

math.MG↗
arXiv2025

The number of trees in distance-hereditary graphs and their friends

Counting the number of spanning trees in specific classes of graphs has attracted increasing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained in the 2020s. The main method is to study the behavior of the vertex (degree) enumerator of a distance-hereditary graph under the operations of copying vertices. Ehrenborg conjecture says that a Ferrer--Young graph maximizes the number of spanning trees among bipartite graphs with the same degree sequence. The second result of this paper is the equivalence of the Ehrenborg conjecture and its polynomial form.

math.CO↗
arXiv2025

Steiner trees with infinitely many terminals on the sides of an angle

The Euclidean Steiner problem is the problem of finding a set $St$, with the shortest length, such that $St \cup A$ is connected, where $A$ is a given set in a Euclidean space. The solutions $St$ to the Steiner problem will be called Steiner sets while the set $A$ will be called input. Since every Steiner set is acyclic we call it Steiner tree in the case when it is connected. We say that a Steiner tree is indecomposable if it does not contain any Steiner tree for a subset of the input. We are interested in finding the Steiner set when the input consists of infinitely many points distributed on two lines. In particular we would like to find a configuration which gives an indecomposable Steiner tree. We consider a self-similar input, namely the set $A_{α,λ}$ of points with coordinates $(λ^{k-1}\cos α,$ $\pm λ^{k-1}\sin α)$, where $λ>0$ and $α>0$ are small fixed values. These points are distributed on the two sides of an angle of size $2α$ in such a way that the distances from the points to the vertex of the angle are in a geometric progression. To our surprise, we show that in this case the solutions to the Steiner problem for $A_{α,λ}$, when $α$ and $λ$ are small enough, are always decomposable trees. More precisely, any Steiner tree for $A_{α,λ}$ is a countable union of Steiner trees, each one connecting 5 points from the input. By considering only a finite number of components we obtain many solutions to the Steiner problem for finite sets composed of $4k+1$ points distributed on the two lines ($2k+1$ on a line and $2k$ on the other line). These solutions are very similar to the ladders of Chung and Graham.

math.MG↗
arXiv2025

An overview of maximal distance minimizers problem

Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $Σ$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, Σ), \] where $dist$ stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its $Γ$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.

math.MG↗
arXiv2024

On set systems without singleton intersections

Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k^2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is $\binom{k^2-k-1}{k-2}$ for every $k > 1$.

math.CO↗
arXiv2023

The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000

We prove that the kissing number in 48 dimensions among antipodal spherical codes with certain forbidden inner products is 52\,416\,000. Constructions of attaining codes as kissing configurations of minimum vectors in even unimodular extremal lattices are well known since the 1970's. We also prove that corresponding spherical 11-designs with the same cardinality are minimal. We use appropriate modifications of the linear programming bounds for spherical codes and designs introduced by Delsarte, Goethals and Seidel in 1977.

math.CO↗
arXiv2023

On stability of weighted spanning tree degree enumerators

Our previous paper shows that the (vertex) spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial (id est is non-zero if all variables have positive imaginary parts) if and only if $G$ is distance-hereditary. In this note we generalize the result on weighted graphs. This generalization allows us to define the class of weighted distance-hereditary graphs.

math.CO↗
arXiv2023

Inverse maximal and average distance minimizer problems

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $Σ$ of the minimal length, such that \[ \max_{y \in M} dist (y, Σ) \leq r. \] The inverse problem is to determine whether a given compact connected set $Σ$ is a minimizer for some compact $M$ and some positive $r$. Let a Steiner tree $St$ with $n$ terminals be unique for its terminal vertices. The first result of the paper is that $St$ is a minimizer for a set $M$ of $n$ points and a small enough positive $r$. It is known that in the planar case a general Steiner tree (on a finite number of terminals) is unique. It is worth noting that a Steiner tree on $n$ terminal vertices can be not a minimizer for any $n$ point set $M$ starting with $n = 4$; the simplest such example is a Steiner tree for the vertices of a square. It is known that a planar maximal distance minimizer is a finite union of simple curves. The second result is an example of a minimizer with an infinite number of corner points (points with two tangent rays which do not belong to the same line), which means that this minimizer can not be represented as a finite union of smooth curves. Our third result is that every injective $C^{1,1}$-curve $Σ$ is a minimizer for a small enough $r>0$ and $M = \overline{B_r(Σ)}$. The proof is based on analogues result by Tilli on average distance minimizers. Finally, we generalize Tilli's result from the plane to $d$-dimensional Euclidean space.

math.MG↗
arXiv2023

On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which the solution of the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations on which at least two locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially require rely upon the theory of subanalytic sets developed in~\cite{bierstone1988semianalytic}. Motivated by this approach we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replace by an arbitrary analytic Riemannian manifold $M$. In this setup we argue that the set of configurations possessing two locally-minimal trees of the same length either has the dimension $n\dim M-1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to abovementioned results, we study the set of set of $n$-point configurations for which there is a unique solution of the Steiner problem in $\mathbb{R}^d$. We show that this set is path-connected.

math.MG↗
arXiv2023

On small non-uniform hypergraphs without property B

For a given hypergraph $H = (V,E)$ consider the sum $q(H)$ of $2^{-|e|}$ over $e \in E$. Consider the class of hypergraphs with the smallest edge of size $n$ and without a 2-colouring without monochromatic edges. Let $q(n)$ be the smallest value of $q(H)$ in this class. We provide a survey of the known bounds on $q(n)$ and make some minor refinements.

math.CO↗