Related papers: Subsquares in random Latin squares and rectangles

The following class of problems arose out of vain attempts to show that the Pascal's triangle adic transformation has trivial spectrum. Partition a set of size $N$ into sets of size $S \equiv S(N)$ (ignoring leftovers). What is the…

We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$…

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n, 1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…

Let X be the random variable that counts the number of triangles in the random graph G(n,p). We show that for some absolute constant c, the probability that X deviates from its expectation by at least \lambda \var(X)^{1/2} is at most…

The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In…

We study conjectures relating degree conditions in $3$-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko…

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…

We study the notion of $\gamma$-negative dependence of random variables. This notion is a relaxation of the notion of negative orthant dependence (which corresponds to $1$-negative dependence), but nevertheless it still ensures…

A partial $(n,k,t)_\lambda$-system is a pair $(X,\mathcal{B})$ where $X$ is an $n$-set of vertices and $\mathcal{B}$ is a collection of $k$-subsets of $X$ called blocks such that each $t$-set of vertices is a subset of at most $\lambda$…

We prove that for each integer $r\geq 2$, there exists a constant $C_r>0$ with the following property: for any $0<\varepsilon \leq 1/2$ and any graph $G$ with clique number at most $r,$ there is a partition of $V(G)$ into at most…

We show that for each positive integer $k$ there is a $k\times k$ matrix $B$ with $\pm 1$ entries such that putting $E$ to be the span of the rows of the $k\times 2k$ matrix $[\sqrt{k}I_k,B]$, then $E,E^{\bot}$ is a Kashin splitting: The…

A multi-graph $G$ on $n$ vertices is $(k,\ell)$-sparse if every subset of $n'\leq n$ vertices spans at most $kn'- \ell$ edges. $G$ is {\em tight} if, in addition, it has exactly $kn - \ell$ edges. For integer values $k$ and $\ell \in [0,…

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements…

In 1962, P\'osa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called…

We study the column subset selection problem with respect to the entrywise $\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{\Omega(1)}$ number of…

What is the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are in a line? This has been asked more than $100$ years ago for $k=2$ and it remained wide open ever since. In this…

Consider an Erd\"os-Renyi random graph in which each edge is present independently with probability 1/2, except for a subset $\sC_N$ of the vertices that form a clique (a completely connected subgraph). We consider the problem of…

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers $k$, there exists a square regular $0,1$ matrix with binary rank $k$, such that the Boolean rank of…

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains…

The chromatic number of a cyclic Latin square of order 2n is 2n+2. The available proof for this statement includes a coloring that is rather lengthy. Here, we introduce a coloring of cyclic Latin square of even order 2n (the Latin square of…