Related papers: Around Sperner's lemma

A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a…

Three central results in economic theory --- Brouwer's fixed-point theorem, Sperner's lemma, and the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma --- are known to be equivalent. In almost all cases, elementary direct proofs of one of these…

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red…

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the…

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random…

Beurling slow variation is generalized to Beurling regular variation. A Uniform Convergence Theorem, not previously known, is proved for those functions of this class that are measurable or have the Baire property. This permits their…

For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old…

In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient…

We consider two CSP problems: the first CSP encodes 2D Sperner's lemma for the standard triangulation of the right triangle on $n^2$ small triangles; the second CSP encodes the fact that it is impossible to match cells of $n \times n$…

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…

In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of…

We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a…

The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $\Gamma$-shape, then the kernel sequence on the top and the kernel sequence at the left have in each position…

In this paper we present a proof of the BMZ Reduction Lemma with a motivational perspective, and state this lemma for maps to manifolds using the classical definition of cohomological dimension. The lemma, proved and utilized in [4], gives…

Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in general position in $d$-space. It is shown that there exist a point $O$ and suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that $|Q_i|\geq…

Let $X,Y$ be finite sets, $r,s,h, \lambda \in \mathbb{N}$ with $s\geq r, X\subsetneq Y$. By $\lambda \binom{X}{h}$ we mean the collection of all $h$-subsets of $X$ where each subset occurs $\lambda$ times. A coloring of…

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…

A celebrated theorem of Spencer states that for every set system $S_1,\dots, S_m \subseteq [n]$, there is a coloring of the ground set with $\{\pm 1\}$ with discrepancy $O(\sqrt{n\log(m/n+2)})$. We provide an algorithm to find such a…

The attempts to prove the Four Color Problem last for long years. A little hope arises that the properties of the minimal partial triangulations will be very useful for the solution of the Four Color Problem. That is why the material of…