arXiv subjects

Shoham Letzter

Publications and source records attributed to Shoham Letzter.

At least 37 records · Page 2Linked to original sources

arXiv2023

The Turán density of tight cycles in three-uniform hypergraphs

The Turán density of an $r$-uniform hypergraph $\mathcal{H}$, denoted $π(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to \infty$. Denote by $\mathcal{C}_{\ell}$ the $3$-uniform tight cycle on $\ell$ vertices. Mubayi and Rödl gave an ``iterated blow-up'' construction showing that the Turán density of $\mathcal{C}_5$ is at least $2\sqrt{3} - 3 \approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain $\mathcal{C}_{\ell}$ for larger $\ell$ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of $\mathcal{C}_{\ell}$ for all large $\ell$ not divisible by $3$, showing that indeed $π(\mathcal{C}_{\ell}) = 2\sqrt{3} - 3$. To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement ``a graph is bipartite if and only if it does not contain an odd cycle''.

math.CO↗

arXiv2023

Ascending Subgraph Decomposition

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph $G$ into copies $H_1, \ldots, H_m$ are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with $\binom{m+1}2$ edges can be decomposed into subgraphs $H_1, \dots, H_m$ such that each $H_i$ has $i$ edges and is isomorphic to a subgraph of $H_{i+1}$. In this paper we prove this conjecture for sufficiently large $m$.

math.CO↗

arXiv2023

Rainbow subdivisions of cliques

We show that for every integer $m \ge 2$ and large $n$, every properly edge-coloured graph on $n$ vertices with at least $n (\log n)^{53}$ edges contains a rainbow subdivision of $K_m$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.

math.CO↗

arXiv2022

Large rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every {\sl proper} colouring of its edges yields a {\sl rainbow} copy of $H$. We study the thresholds for such so-called {\sl anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every $s$. In this paper, we show that for $s \geq 9$ the threshold is $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 \leq s \leq 7$, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$, and show that the threshold for this property is $n^{-2}$ for every $\ell \geq 2$; in particular, it does not depend on the length of the cycle $C_{2\ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

math.CO↗

arXiv2022

Small rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every \emph{proper} colouring of its edges yields a \emph{rainbow} copy of $H$. We study the thresholds for such so-called \emph{anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_\ell$ is $n^{-1/m_2(K_{\left\lceil \ell/2 \right\rceil})}$, whenever $\ell \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for $4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{large} cliques. In particular, we show that the thresholds for $\ell \in \{4, 5, 7\}$ are $n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $\ell \in \{6, 8\}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $\ell = 3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.

math.CO↗

arXiv2022

An improvement on Łuczak's connected matchings method

A connected matching in a graph $G$ is a matching contained in a connected component of $G$. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the $3$-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007).

math.CO↗

arXiv2022

Immersion of complete digraphs in Eulerian digraphs

A digraph $G$ \emph{immerses} a digraph $H$ if there is an injection $f : V(H) \to V(G)$ and a collection of pairwise edge-disjoint directed paths $P_{uv}$, for $uv \in E(H)$, such that $P_{uv}$ starts at $u$ and ends at $v$. We prove that every Eulerian digraph with minimum out-degree $t$ immerses a complete digraph on $Ω(t)$ vertices, thus answering a question of DeVos, Mcdonald, Mohar, and Scheide.

math.CO↗

arXiv2022

Hypergraphs with no tight cycles

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor.

math.CO↗

arXiv2021

The size-Ramsey number of 3-uniform tight paths

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rödl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.

math.CO↗

arXiv2021

Size-Ramsey numbers of powers of hypergraph trees and long subdivisions

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends and strongly generalises recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

math.CO↗

arXiv2020

Cycle partitions of regular graphs

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = Ω(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = Ω(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for bipartite graphs).

math.CO↗

arXiv2020

Monochromatic triangle packings in red-blue graphs

We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erdős. We also prove a corresponding stability result, showing that $2$-colourings that are close to attaining the aforementioned bound have a colour class which is close to bipartite. As part of our proof, we confirm a recent conjecture of Tyomkyn about the fractional version of this problem.

math.CO↗

arXiv2020

Fractional triangle decompositions in almost complete graphs

We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured complete graph contains $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, which confirms a conjecture of Erdős.

math.CO↗

arXiv2020

Minimum degree conditions for monochromatic cycle partitioning

A classical result of Erdős, Gyárfás and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.

math.CO↗

arXiv2020

Lagrangians of Hypergraphs II: When colex is best

A well-known conjecture of Frankl and Füredi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona.

math.CO↗

arXiv2020

Hypergraph Lagrangians I: the Frankl-Füredi conjecture is false

An old and well-known conjecture of Frankl and Füredi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r \ge 4$. We also show that, for sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range $\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.

math.CO↗

arXiv2020

Orthonormal representations of $H$-free graphs

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any three there is an orthogonal pair. How large can $n$ be as a function of $d$, and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these two celebrated questions, asked by Erdős and Lovász, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovász $\vartheta$-function and minimum semidefinite rank. In this paper, we study these parameters for general $H$-free graphs. In particular, we show that for certain bipartite graphs $H$, there is a connection between the Turán number of $H$ and the maximum of $\vartheta \left( \overline{G} \right)$ over all $H$-free graphs $G$.

math.CO↗

arXiv2019

Optimal Adaptive Detection of Monotone Patterns

We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon [n] \to \mathbb{R}$, provided that $f$ is $\varepsilon$-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes $O(\log n)$ queries. This is optimal, matching the classical $Ω(\log n)$ adaptive lower bound by Fischer [2004] for monotonicity testing (which corresponds to the case $k=2$), and implying in particular that the query complexity of testing whether the longest increasing subsequence (LIS) has constant length is $Θ(\log n)$.

cs.DS↗