arXiv subjects

Mikhail V. Bludov

Publications and source records attributed to Mikhail V. Bludov.

6 recordsLinked to original sources

arXiv2026

Enumerating Minimal Balanced Collections

In this note, we explore the combinatorics of balanced collections. A collection of subsets of the set $[n] = \{1, \dots, n\}$ is called \emph{balanced} if the relative interior of the convex hull of the corresponding characteristic vectors intersects the main diagonal of the $n$-dimensional cube at a point other than the origin, and it is called \emph{minimal} if it contains no proper balanced subcollections. We determine the asymptotic number of minimal balanced collections. Specifically, if $B_n$ denotes their total number, then $B_n=2^{n^2-n+1}\bigl(1+o(1)\bigr)/n!$ as $n\to\infty$. We discuss applications of this result to fractional matchings in hypergraphs and to resonance arrangement

math.CO↗

arXiv2026

Complexes of C-Unbalanced Subsets and Balanced-Subset Posets

Let $V=\{v_1,\dots,v_m\}\subset\mathbb{R}^d$. We study simplicial complexes arising from subsets of $V$ whose convex hulls avoid a prescribed convex set. Given a convex set $C\subset\mathbb{R}^d$, let $\mathcal K(V,C)$ be the simplicial complex consisting of all subsets $S\subset V$ such that $\operatorname{conv}(S)\cap C=\emptyset$. We prove that $\mathcal K(V,C)$ is homotopy equivalent to the union of the faces of $\operatorname{conv}(V)$ that are disjoint from $C$. In the special case $C=\{r\}$, subsets $S$ satisfying $r\in\operatorname{conv}(S)$ are called weakly $r$-balanced. If $r\in\operatorname{relint}\operatorname{conv}(V)$ and the poset of proper weakly $r$-balanced subsets is nonempty, then its order complex is homotopy equivalent to the sphere $S^{m-k-2}$, where $k=\dim\operatorname{aff}(V)$.

math.CO↗

arXiv2026

On Scarf's theorem for Generalized Cooperative Games

In this paper, we study a generalization of cooperative games with non-transferable utility. In our model, coalitions are replaced by firms: each firm is assigned a resource vector, while the set of utility vectors of a coalition is replaced by a general comprehensive set of feasible payoff vectors of this firm in a common payoff space. We introduce the notions of core and fractional core for such games and relate their existence to homotopy invariants of covers associated with the game. The main result shows that the fractional core is nonempty precisely when the associated cover is homotopicaly nontrivial. As a consequence, we obtain a Scarf-type theorem for generalized cooperative games.

math.CO↗

arXiv2025

On essential simplicial maps $S^3 \rightarrow S^2$

A fiber-uniform bound on the complexity of an essential simplicial map $S^3\rightarrow S^2$ is proven, and the tightness of the bound is investigated. It follows that the triangulation of the Hopf map constructed by Madahar and Sarkaria is minimal in its homotopy class in terms of the number of 3-simplices in the triangulation of $S^3$.

math.AT↗

arXiv2025

Balanced sets and homotopy invariants of covers

In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of their convex hull. We show that the simplicial complex of non-balanced subsets associated with $(V,r)$ has the homotopy type of a sphere, and use this to develop a theory of homotopy invariants of covers relative to balanced sets. A key result is that the homotopy class of a cover depends only, up to an involution, on the balanced-equivalence class of $(V,r)$. As applications, we obtain extension theorems for covers in this setting and derive the KKMS lemma, its analogues, and related combinatorial fixed-point results.

math.CO↗

arXiv2023

Balanced 2-subsets

Balanced sets appeared in the 1960s in cooperative game theory as a part of nonempty core conditions. In this paper we present a classification of balanced families containing only 2-element subsets. We also discuss generalizations of the classical Sperner and Tucker lemmas using balanced sets.

math.CO↗