Related papers: An unavoidable set of D-reducible configurations

This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Gr\"unbaum colorings, their existence and usage. In particular, one of the most…

Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in…

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of…

We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials),…

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

We give a pictorial proof that transparently illustrates why four colours suffce to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show,…

We introduce a new invariant of a cubic graph - its regular colouring defect - which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a…

Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated…

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset…

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of…

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for…

We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which $d$-dimensional configurations and finite patterns are presented as formal…

Say a graph $G$ is a {\em pentagraph} if every cycle has length at least five, and every induced cycle of odd length has length five. N. Robertson proposed the conjecture that the Petersen graph is the only pentagraph that is…

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…

Let $(E,V)$ be a general generated coherent system of type $(n,d,n+m)$ on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of $E$ to the semistability of the kernel of the…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $\Delta(G)$ of $G$. The only case that the conjecture is resolved is…

Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are…