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If you color a table using k colors, and throw a needle randomly on it, for some proper definition, you get a certain probability that the endpoints will fall on different colors. How can one make this probability maximal? This problem is…
The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…
A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…
In 1970 Hajnal and Szemer\'edi proved a conjecture of Erd\"os that for a graph with maximum degree $\Delta$, there exists an equitable $\Delta+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand…
For fixed integers $p$ and $q$, let $f(n,p,q)$ denote the minimum number of colors needed to color all of the edges of the complete graph $K_n$ such that no clique of $p$ vertices spans fewer than $q$ distinct colors. Any edge-coloring with…
One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy…
Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…
In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree admits a Borel unfriendly coloring…
We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the…
We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…
Colored graphical models provide a parsimonious approach to modeling high-dimensional data by exploiting symmetries in the model parameters. In this work, we introduce the notion of coloring for extremal graphical models on multivariate…
The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey…
An algorithm is presented that, taking a sequence of independent Bernoulli random variables with parameter $1/2$ as inputs and using only rational arithmetic, simulates a Bernoulli random variable with possibly irrational parameter $\tau$.…
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the…
Probabilistic zero-forcing is a coloring process on a graph. In this process, an initial set of vertices is colored blue, and the remaining vertices are colored white. At each time step, blue vertices have a non-zero probability of forcing…
In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that…
In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent…
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property.…
We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges…
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number…