Matias Pavez-Signé

University of Chile

We show that for every $n\in \mathbb N$ and $\log n\le d\lt n$ , if a graph $G$ has $N=\Theta (dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$ , then it contains a spanning subdivision of every $n$ -vertex $d$ -regular graph.

Let $R(G)$ be the $2$-colour Ramsey number of a graph $G$.In this note, we prove that for any non-decreasing function $n \leq f(n) \leq R(K_n)$, there exists a sequence of connected graphs $(G_n)_{n\in\mathbb N}$, with $|V(G_n)| = n$ for all $n \geq 1$, such that $R(G_n) = \Theta(f(n))$. In contrast, we also show that an analogous statement does no...

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \...

We show that for every $\Delta\in\mathbb N$, there exists a constant $C$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda\ge C$ and $d$ is large enough, then $G^2$ contains every $n$-vertex tree with maximum degree bounded by $\Delta$. This answers a question of Krivelevich.

We show that for every $n\in\mathbb N$ and $\log n\le d\le n$, if a graph $G$ has $N=\Theta(dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.

For graphs G,H$$ G,H $$ and a family of graphs ℱ$$ \mathcal{F} $$, we write G→(H,ℱ)$$ G\to \left(H,\mathcal{F}\right) $$ to denote that every blue‐red coloring of the edges of G$$ G $$ contains either a blue copy of H$$ H $$, or a red copy of each F∈ℱ$$ F\in \mathcal{F} $$. For integers n$$ n $$ and D$$ D $$, let 𝒯(n,D) denote the family of all tre...

Let $R(G)$ be the two-colour Ramsey number of a graph $G$. In this note, we prove that for any non-decreasing function $n \leq f(n) \leq R(K_n)$, there exists a sequence of connected graphs $(G_n)_{n\in\mathbb N}$, with $|V(G_n)| = n$ for all $n \geq 1$, such that $R(G_n) = \Theta(f(n))$. In contrast, we also show that an analogous statement does n...

Let $R(C_n)$ be the Ramsey number of the cycle on $n$ vertices. We prove that, for some $C > 0$, with high probability every $2$-colouring of the edges of $G(N,p)$ has a monochromatic copy of $C_n$, as long as $N\geq R(C_n) + C/p$ and $p \geq C/n$. This is sharp up to the value of $C$ and it improves results of Letzter and of Krivelevich, Kronenber...

Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung–Graham–Wilson theorem for graphs we provide a list of wor...

A word on q symbols is a sequence of letters from a fixed alphabet of size q. For an integer k⩾1, we say that a word w is k-universal if, given an arbitrary word of length k, one can obtain it by removing letters from w. It is easily seen that the minimum length of a k-universal word on q symbols is exactly qk. We prove that almost every word of si...

We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the P\'osa--Seymour conjecture. Moreover, we prove that...

We prove that for fixed k, every k-uniform hypergraph on n vertices and of minimum codegree at least n/2+o(n) contains every spanning tight k-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp .

We prove the Erdős–Sós conjecture for trees with bounded maximum degree and large dense host graphs. As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant.

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is asymptotically sharp. We also prove an extension...

Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung-Graham-Wilson theorem for graphs we provide a list of wor...

Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung-Graham-Wilson theorem for graphs we provide a list of wor...

A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols is exactly $qk$. We prove that...

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every red-blue colouring of the edges of $G$ contains either a blue $K_{r+1}$, or red copies of every tree with $n$ edges and maximum degree at most $D$. In 1977, Chv\'atal proved that, for any integers $r,n,D \ge 2$, $K_N \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$...

We prove the Erd\H os--S\'os conjecture for trees with bounded maximum degree and large dense host graphs.

We propose the following conjecture: For every fixed $\alpha\in [0,\frac 12]$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. \\ Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We...

We conjecture that every n-vertex graph of minimum degree at least k 2 and maximum degree at least 2k contains all trees with k edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd˝ os-Sós conjecture and the 2 3-conjecture. We prove an app...

We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd\H os--S\'os conjecture and the $\frac 23$-con...

We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps in Banach spaces. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point....