Related papers: Finding a non-minority ball with majority answers

We address the problem of computing the minimum number of triangles to separate a set of blue points from a set of red points in $\mathbb{R}^2$. A set of triangles is a \emph{separator} of one color from the other if every point of that…

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer…

We study a P\'olya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with…

We study a quantitative Ramsey-type problem on 3-term arithmetic progressions: how should the set of integers $[n] = \{1, 2, \dots, n\}$ be colored using 3 colors in order to maximize the number of rainbow 3-term arithmetic progressions? By…

A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as…

In this paper we consider clustering problems in which each point is endowed with a color. The goal is to cluster the points to minimize the classical clustering cost but with the additional constraint that no color is over-represented in…

We play a variation of Nim on stacks of tokens. Take your favorite increasing sequence of positive integers and color the tokens according to the following rule. Each token on a level that corresponds to a number in the sequence is colored…

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…

For a ring R and system L of linear homogeneous equations, we call a coloring of the nonzero elements of R minimal for L if there are no monochromatic solutions to L and the coloring uses as few colors as possible. For a rational number q…

The problem of 2-coloring uniform hypergraphs has been extensively studied over the last few decades. An n-uniform hypergraph is not 2-colorable if its vertices can't be colored with two colors, Red and Blue, such that every hyperedge…

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and…

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform…

Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class.…

In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed…

A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of…

Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $\alpha$ balls of the same colour and $\beta$ balls of the opposite colour. Both cases, $\beta=0$ and $\beta>0$ are well…

In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects…

Consider the following process whereby $n$ balls are distributed into $k$ bins. Repeatedly, a ball is removed from a non-empty bin chosen uniformly at random. The process ends when a single non-empty bin remains. Will Ma…

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together…