Related papers: Scarf's theorems, simplices, and oriented matroids
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to…
A coloring of a matroid is an assignment of colors to the elements of its ground set. We restrict to proper colorings - those for which elements of the same color form an independent set. Seymour proved that a $k$-colorable matroid is also…
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 $n \times n$…
A new theoretical technique for understanding, analyzing and developing optical systems is presented. The approach is statistical in nature, where information about an object under investigation is discovered, by examining deviations from a…
We consider Brouwer's fixed point theorem and Sperner's lemma in one dimension. We present a proof of the Brouwer theorem using the Sperner lemma, and vice versa. However, we also show that they are not equivalent, because the Sperner lemma…
We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated…
A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows…
This paper provides a one-line proof of Frequent Directions (FD) for sketching streams of matrices. The simpler proof arises from sketching the covariance of the stream of matrices rather than the stream itself.
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many…
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{\"o}nflies theorem.…
A (finite, undirected) graph is $(n,k)$-colourable if we can assign each vertex a $k$-subset of $\{1,2,\ldots,n\}$ so that adjacent vertices receive disjoint subsets. We consider the following problem: if a graph is $(n,k)$-colourable, then…
We present a proof of Arrow's theorem from social choice theory that uses a fixpoint argument. Specifically, we use Banach's result on the existence of a fixpoint of a contractive map defined on a complete metric space. Conceptually, our…
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a…
Allegedly, Brouwer discovered his famous fixed point theorem while stirring a cup of coffee and noticing that there is always at least one point in the liquid that does not move. In this paper, based on a talk in honour of Brouwer at the…
In this note, we revisit the notion of strong set-colorings introduced by Hegde (2009) and completed by equivalences due to Boutin et al. (2010) and provide a necessary and sufficient \emph{Steiner packing} characterisation: a finite graph…
We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of…
We show that the map from $K({\mathbb S})$ to its chromatic completion is a connective cover and identify the fiber in $K$-theoretic terms. We combine this with recent work of Land-Mathew-Meier-Tamme to prove a form of "Waldhausen's…
Menger's Theorem is a fundamental result in graph theory. It states that if in a graph $G$ with distinguished sets of terminal vertices $S$ and $T$ there are no $k$ pairwise vertex-disjoint $S$-$T$ paths, then there is a set of less than…
We prove that Lefschetz's principle of approximating the cohomology of a possibly singular affine scheme of finite type over a field by the cohomology of a suitable (thickening of a) hyperplane section can be made uniform: in the affine…