Related papers: Exponential Erd\H{o}s-Szekeres theorem for matrice…

Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erd\H{o}s-Burgess constant} of $\mathcal{S}$ is defined as the…

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and…

In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of…

Let $\Sigma=\{a_1, \ldots , a_n\}$ be a set of positive integers with $a_1 < \ldots < a_n$ such that all $2^n$ subset sums are pairwise distinct. A famous conjecture of Erd\H{o}s states that $a_n>C\cdot 2^n$ for some constant $C$, while the…

In 1965, Paul Erd\H{o}s asked about the largest family $Y$ of $k$-sets in $\{ 1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erd\H{o}s Matching Conjecture. We investigate…

The famous Brown-Erd\H{o}s-S\'os conjecture from 1973 states, in an equivalent form, that for any fixed $\delta>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges…

We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental…

We show a general result known as the Erdos_Sos Conjecture: if $E(G)>{1/2}(k-1)n$ where $G$ has order $n$ then $G$ contains every tree of order $k+1$ as a subgraph.

E. B. Davies et B. Simon have shown (among other things) the following result: if T is an n\times n matrix such that its spectrum \sigma(T) is included in the open unit disc \mathbb{D}=\{z\in\mathbb{C}:\,|z|<1\} and if…

Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $1\leq r\leq m\leq n$ and $\mathbb{Z}_{h}^{m\times n}$ be the set of all…

In 1962, Erd\H{o}s proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $\delta(G)\geq k$…

The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of…

In 1994, Erd\H{o}s, Gy\'{a}rf\'{a}s and {\L}uczak posed the following problem: given disjoint vertex sets $V_1,\dots,V_n$ of size~$k$, with exactly one edge between any pair $V_i,V_j$, how large can $n$ be such that there will always be an…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

Erd\H{o}s and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erd\H{o}s-Hajnal…

Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the…

Fillmore Theorem says that if A is an nxn complex non-scalar matrix and {\gamma}_1,...,{\gamma}_{n} are complex numbers with {\gamma}_1+...+{\gamma}_{n}=trA, then there exists a matrix B similar to A with diagonal entries…

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows.…

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most…

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. For $n\geq 1$ set $G_n = g_n \ldots g_1$. Given any starting point $x=\mathbb R v\in\mathbb{P}^{d-1}$, consider the…