arXiv subjects

Matías Pavez-Signé

Publications and source records attributed to Matías Pavez-Signé.

At least 19 recordsLinked to original sources

arXiv2026

Spanning clique subdivisions in pseudorandom graphs

In this paper, we study the appearance of a spanning subdivision of a clique in graphs satisfying certain pseudorandom conditions. Specifically, we show the following three results. Firstly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/λ\ge C$, every $(n,d,λ)$-graph contains a spanning subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log n}}\}$. Secondly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/λ\ge C\log^3n$, every $(n,d,λ)$-graph contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log^3n}}\}$. Finally, we show that for every $μ>0$, there are constants $c,\varepsilon\in (0,1]$ and $n_0\in \mathbb N$ such that, whenever $n\ge n_0$, every $n$-vertex graph with minimum degree at least $μn$ and no bipartite holes of size $\varepsilon n$ contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le c\sqrt{n}$.

math.CO↗

arXiv2026

A degree version of the Burr-Erdős conjecture on trees

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $Δ\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $δ(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $Δ$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

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arXiv2026

Ramsey numbers for 1-degenerate 3-graphs

We construct a 3-uniform 1-degenerate hypergraph on $n$ vertices whose 2-colour Ramsey number is $Ω\big(n^{3/2}/\log n\big)$. This shows that all remaining open cases of the hypergraph Burr-Erdős conjecture are false. Our graph is a variant of the celebrated hedgehog graph. We additionally show near-sharp upper bounds, proving that all 3-uniform generalised hedgehogs have 2-colour Ramsey number $O\big(n^{3/2}\big)$.

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arXiv2026

The Lovász conjecture holds for moderately dense Cayley graphs

We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lovász conjecture and improves upon the previous best result of this form due to Christofides, Hladký, and Máthé from 2014 concerning graphs with $d\geq \varepsilon n$. Our proof avoids the use of Szemerédi's regularity lemma and relies instead on an efficient arithmetic regularity lemma specialised to Cayley graphs.

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arXiv2025

Ramsey numbers of trees

We show that there exists a constant $c>0$ such that every $n$-vertex tree $T$ with $Δ(T)\le cn$ has Ramsey number $R(T)=\max\{t_1+2t_2,2t_1\}-1$, where $t_1\ge t_2$ are the sizes of the bipartition classes of $T$. This improves an asymptotic result of Haxell, Łuczak, and Tingley from 2002, and shows that, though Burr's 1974 conjecture on the Ramsey numbers of trees has long been known to be false for certain `double stars', it is true for trees with up to small linear maximum degree.

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arXiv2025

Hamilton cycles in pseudorandom graphs: resilience and approximate decompositions

Dirac's classical theorem asserts that, for $n \ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian. Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba, Kühn, Lo, Osthus and Treglown, they admit a decomposition into Hamilton cycles and at most one perfect matching, solving the well-known Nash-Williams conjecture. In the pseudorandom setting, it has long been conjectured that similar results hold in much sparser graphs. We prove two overarching theorems for graphs that exclude excessively dense subgraphs, which yield asymptotically optimal resilience and Hamilton-decomposition results in sparse pseudorandom graphs. In particular, our results imply that for every fixed $γ> 0$, there exists a constant $C > 0$ such that if $G$ is a spanning subgraph of an $(n,d,λ)$-graph satisfying $δ(G) \ge (\tfrac12 + γ)d$ and $d/λ\ge C$, then $G$ must contain a Hamilton cycle. Secondly, we show that for every $\varepsilon > 0$, there is $C > 0$ so that every $(n,d,λ)$-graph with $d/λ\ge C$ contains at least $(\tfrac12 - \varepsilon)d$ edge-disjoint Hamilton cycles, and, finally, we prove that the entire edge set of $G$ can be covered by no more than $(\tfrac12 + \varepsilon)d$ such cycles. All bounds are asymptotically optimal and significantly improve earlier results on Hamiltonian resilience, packing, and covering in sparse pseudorandom graphs.

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arXiv2025

Colour-bias perfect matchings in hypergraphs

We study conditions under which an edge-coloured hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock and Sgueglia, and of Balogh, Treglown and Zárate-Guerén.

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arXiv2024

Ramsey numbers of cycles in random graphs

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, Kronenberg and Mond.

math.CO↗

arXiv2024

Monochromatic partitions in 2-edge-coloured bipartite graphs

We study two variations of the Gyarfas--Lehel conjecture on the minimum number of monochromatic components needed to cover an edge-coloured complete bipartite graph. Specifically, we show the following. - For p>> (\log n/n)^{1/2}, w.h.p.~every 2-colouring of the random bipartite graph G~ G(n,n,p) admits a cover of all but O(1/p) vertices of G using at most three vertex-disjoint monochromatic components. - For every 2-colouring of a bipartite graph G with parts of size n and minimum degree (13/16+o(1))n, the vertices of G can be covered using at most three vertex-disjoint monochromatic components.

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arXiv2023

Spanning trees in the square of pseudorandom graphs

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

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arXiv2023

Spanning trees in pseudorandom graphs via sorting networks

We show that $(n,d,λ)$-graphs with $λ=O(d/\log^3 n)$ are universal with respect to all bounded degree spanning trees. This significantly improves upon the previous best bound due to Han and Yang of the form $λ=d/\exp{(O(\sqrt{\log n}))}$, and makes progress towards a problem of Alon, Krivelevich, and Sudakov from 2007. Our proof relies on the existence of sorting networks of logarithmic depth, as given by a celebrated construction of Ajtai, Komlós and Szemerédi. Using this construction, we show that the classical vertex-disjoint paths problem can be solved for a set of vertices fixed in advance.

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arXiv2023

Ramsey numbers of bounded degree trees versus general graphs

For every $k\ge 2$ and $Δ$, we prove that there exists a constant $C_{Δ,k}$ such that the following holds. For every graph $H$ with $χ(H)=k$ and every tree with at least $C_{Δ,k}|H|$ vertices and maximum degree at most $Δ$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+σ(H)$, where $σ(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{Δ,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

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arXiv2023

Ramsey numbers with prescribed rate of growth

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 $R(G_n) = Θ(f(n))$. In contrast, we also show that an analogous statement does not hold for hypergraphs of uniformity at least $5$. We also use our techniques to answer a question posed by DeBiasio about the existence of sequences of graphs whose $2$-colour Ramsey number is linear whereas their $3$-colour Ramsey number has superlinear growth.

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arXiv2023

Spanning subdivisions in Dirac graphs

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

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arXiv2023

Counting spanning subgraphs in dense hypergraphs

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 $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp(n\log n-Θ(n))$ Hamilton $\ell$-cycles as long as $(k-\ell)\mid n$. When $(k-\ell)\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, Kühn and Osthus, while, when $(k-\ell)\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell<k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

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arXiv2023

Towards a hypergraph version of the Pósa-Seymour conjecture

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 Pósa-Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than $r$ which admits a tree decomposition where every vertex is in a bounded number of bags.

math.CO↗

arXiv2023

Dirac-type conditions for spanning bounded-degree hypertrees

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 sub\-graph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.

math.CO↗

arXiv2021

Quasi-random words and limits of word sequences

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 word properties which are equivalent to uniformity. In particular, we show that uniformity is equivalent to counting 3-letter subsequences. Inspired by graph limit theory we then investigate limits of convergent word sequences, those in which all subsequence densities converge. We show that convergent word sequences have a natural limit, namely Lebesgue measurable functions of the form $f:[0,1]\to[0,1]$. Via this theory we show that every hereditary word property is testable, address the problem of finite forcibility for word limits and establish as a byproduct a new model of random word sequences. Along the lines of the proof of the existence of word limits, we can also establish the existence of limits for higher dimensional structures. In particular, we obtain an alternative proof of the result by Hoppen, Kohayakawa, Moreira, Ráth and Sampaio [{\it J. Combin. Theory Ser. B 103(1):93--113, 2013}] establishing the existence of permutons.

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