Related papers: Cutting a part from many measures

A well known generalization of Alon's "splitting nacklace theorem" by Longueville and Zivaljevic states that every k-colored n-dimensional cube can be fairly split using only k cuts in each dimension. Here we prove that for every t there…

Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A' \subset A$ and $B' \subset B$. We say that such a partition has the…

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 cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions…

Karasev conjectured that for any set of $3k$ lines in general position in the plane, which is partitioned into $3$ color classes of equal size $k$, the set can be partitioned into $k$ colorful 3-subsets such that all the triangles formed by…

Tverberg's theorem states that for any $k \ge 2$ and any set $P \subset \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points in $d$ dimensions, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The…

It is proved that if we partition a $d$-dimensional cube into $n^d$ small cubes and color the small cubes into $m+1$ colors then there exists a monochromatic connected component consisting of at least $f(d, m) n^{d-m}$ small cubes.

Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that…

The coloured Tverberg theorem was conjectured by B\'ar\'any, Lov\'{a}sz and F\"uredi and asks whether for any d+1 sets (considered as colour classes) of k points each in R^d there is a partition of them into k colourful sets whose convex…

We will prove the following generalization of the ham sandwich Theorem, conjectured by Imre B\'ar\'any. Given a positive integer $k$ and $d$ nice measures $\mu_1, \mu_2,..., \mu_d$ in $\mathbb{R}^d$ such that $\mu_i (\mathds{R}^d) = k$ for…

A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by…

Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

Given integers $m\le c$ and an exact $c$-coloring of the edges of a complete countably infinite graph (i.e. a coloring that uses exactly $c$ colors), must there be an infinite subgraph that is exactly $m$-colored? Using the Infinite Ramsey…

We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing…

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…

Let $b^{k}_{\ell,m}(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\ell$ or $m$. We establish several congruence relations for $b_{\ell,m}(n)$. For instance, for any nonnegative integer $n$…

It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller…

Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be…

Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in…