arXiv subjects

Matías Pavez-Signé

Publications and source records attributed to Matías Pavez-Signé.

24 records · Page 2Linked to original sources

arXiv2021

Universal arrays

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 almost every word of size $(1+o(1))c_qk$ is $k$-universal with high probability, where $c_q$ is an explicit constant whose value is roughly $q\log q$. Moreover, we show that the $k$-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no. 1, 1--32], we give asymptotically tight bounds for every higher dimensional analogue of this problem.

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arXiv2021

Ramsey goodness of trees in random graphs

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$. In 1977, Chvátal proved that for any integers $r,n,D \ge 2$, $K_N \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if and only if $N \ge rn+1$. We prove a random analogue of Chvátal's theorem for bounded degree trees, that is, we show that for each $r,D\ge 2$ there exist constants $C,C'>0$ such that if $p \ge C{n}^{-2/(r+2)}$ and $N \geq rn + C'/p$, then \[G(N,p) \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)\] with high probability as $n\to \infty$. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős--Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.

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arXiv2020

Maximum and minimum degree conditions for embedding trees

We propose the following conjecture: For every fixed $α\in [0,\frac 13)$, each graph of minimum degree at least $(1+α)\frac k2$ and maximum degree at least $2(1-α)k$ 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 also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding $\frac k2$ and $\frac 43k$, respectively, as long as the host graph avoids a specific structure.

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arXiv2020

On the Erdős-Sós conjecture for trees with bounded degree

We prove the Erd\H os--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.

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arXiv2018

Degree conditions for embedding trees

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--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.

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arXiv2017

Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces

We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. 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. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.

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