arXiv subjects
Publications and source records attributed to Rajko Nenadov.
In 1986, Füredi and Kahn showed that the dimension $\dim(P)$ of any finite poset $P$ satisfies $\dim(P) = O(d \log^2 d)$, where $d$ is the maximum degree of the comparability graph of $P$. Scott and Wood more recently improved this bound to one of the form $d \log^{1+o(1)} d$. We show that $\dim(P) = O(d \log d)$, thus confirming that the corresponding lower bound of Erdős, Kierstead, and Trotter is tight up to the implicit constant.
In graph bootstrap percolation, edges of an Erdős-Rényi random graph ${\mathcal G}_{n,p}$ are initially active, and activation spreads to other edges of $K_n$ via the combinatorics of a fixed graph $H$: an edge becomes active whenever it is the unique inactive edge in a copy of $H$. The process $H$-percolates if all edges of $K_n$ are eventually activated. While classical cases such as $H=K_3$ (connectivity) and $H=K_4$ (related to $2$-neighbor bootstrap percolation) have been studied extensively, general graphs $H$ can exhibit wildly different behaviors. In this work, we determine the critical $H$-percolation threshold $p_c(n,H)$ for every graph $H$, fully resolving a longstanding open question of Balogh, Bollobás, and Morris. The location of $p_c(n,H)$ is governed by a new, universal parameter $ρ(H)$, which measures the maximal efficiency of witness graphs that activate an edge. To achieve this, we introduce a novel framework based on the unfolding and refolding of witness graphs. While previous works were restricted to specific families of $H$, our approach provides a unified strategy for all $H$. Inspired by algebraic topology, we lift witness graphs to covering graphs and algorithmically embed folded versions into ${\mathcal G}_{n,p}$ via a sequence of extensions. Crucially, this allows us to incorporate highly efficient witness graphs of unbounded size, which are potentially far larger than ${\mathcal G}_{n,p}$ itself. Beyond resolving $p_c(n,H)$, our framework recovers and strengthens several existing bounds in the literature. Finally, we initiate the study of the universal density parameter $ρ(H)$ and pose central open questions regarding its computability and its exact correspondence with the sharpness of the $H$-percolation threshold.
We show that there is a constant $C>0$ such that for each integer $n\geq 1$, there is a poset on at most $2^{2n/3+C\sqrt{n}}$ elements that contains each $n$-element poset as an (induced) subposet.
Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. The proof is based on a bespoke container argument.
This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse $d$-regular \emph{random} graphs, in the above proof systems, with high probability requires proofs with degree $Ω(n/\log n)$. We extend their result by showing the same lower bound holds for \emph{all} $d$-regular graphs with a mild spectral gap.
The cycle set of a graph $G$ is the set consisting of all sizes of cycles in $G$. Answering a conjecture of Erdős and Faudree, Verstraëte showed that there are at most $2^{n - n^{1/10}}$ different cycle sets of graphs with $n$ vertices. We improve this bound to $2^{n - n^{1/2 - o(1)}}$. Our proof follows the general strategy of Verstraëte of reducing the problem to counting cycle sets of Hamiltonian graphs with many chords or a large maximum degree. The key new ingredients are near-optimal container lemmata for cycle sets of such graphs.
We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $αn / t$ expands by a factor of $t$, for some (constant) $α> 0$ and large $t = t(n)$. We obtain the following: * Improving results of Krivelevich and Sudakov, we show that a small-set expander contains a complete minor of order $\sqrt{n t / \log n}$. * We show that a small-set expander contains every graph $H$ with $O(n \log t / \log n)$ edges and vertices as a minor. We complement this with an upper bound showing that if an $n$-vertex graph $G$ has average degree $d$, then there exists a graph with $O(n \log d / \log n)$ edges and vertices which is not a minor of $G$. This has two consequences: (i) It implies the optimality of our result in the case $t = d^c$ for some constant $c > 0$, and (ii) it shows expanders are optimal minor-universal graphs of a given average degree.
For a field $\mathbb{F}$ and integers $d, k$ and $\ell$, a set $A \subseteq \mathbb{F}^d$ is called $(k,\ell)$-nearly orthogonal if all vectors in $A$ are non-self-orthogonal and every $k+1$ vectors in $A$ contain $\ell + 1$ pairwise orthogonal vectors. Recently, Haviv, Mattheus, Milojević and Wigderson have improved the lower bound on nearly orthogonal sets over finite fields, using counting arguments and a hypergraph container lemma. They showed that for every prime $p$ and an integer $\ell$, there is a constant $δ(p,\ell)$ such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $d \geq k \geq \ell + 1$, $\mathbb{F}^d$ contains a $(k,\ell)$-nearly orthogonal set of size $d^{δk / \log k}$. This nearly matches an upper bound $\binom{d+k}{k}$ coming from Ramsey theory. Moreover, they proved the same lower bound for the size of a largest set $A$ where for any two subsets of $A$ of size $k+1$ each, there is a vector in one of the subsets orthogonal to a vector in the other one. We prove a common generalisation of this result, showing that essentially the same lower bound holds for the size of a largest set $A \subseteq \mathbb{F}^d$ with the stronger property that given any family of subsets $A_1, \ldots, A_{\ell+1} \subseteq A$, each of size $k+1$, we can find a vector in each $A_i$ such that they are all pairwise orthogonal. Rather than combining both counting and container arguments, we make use of a multipartite asymmetric container lemma that allows for non-uniform co-degree conditions. This lemma was first discovered by Campos, Coulson, Serra and Wötzel, and we provide a new and short proof for this lemma.
The Cayley sum graph $Γ_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if $S$ is formed by taking each element in $\mathbb{Z}_n$ independently with probability $p$, for $p > (\log n)^{-1/80}$, then with high probability the largest independent set in $Γ_S$ is of size $$ (2 + o(1)) \log_{1/(1-p)}(n). $$ This extends a result of Green and Morris, who considered the case $p = 1/2$, and asymptotically matches the independence number of the binomial random graph $G(n,p)$. We improve the range of $p$ for which this holds to $p > (\log n)^{-1/3 + o(1)}$. The heavy lifting has been done by Campos, Dahia, and Marciano, and we show that their key lemma can be used a bit more efficiently.
We investigate the problem of reconstructing a set $P\subseteq \mathbb{R}$ of distinct points, where the only information available about $P$ consists of the distances between some of the pairs of points. More precisely, we examine which properties of the graph $G$ of known distances, defined on the vertex set $P$, ensure that $P$ can be uniquely reconstructed up to isometry. We prove that as soon as the random graph process has minimum degree 2, with high probability it can reconstruct all distances within any point set in $\mathbb{R}$. This resolves a conjecture of Benjamini and Tzalik. We also study the feasibility and limitations of reconstructing the distances within almost all points using much sparser random graphs. In doing so, we resolve a question posed by Girão, Illingworth, Michel, Powierski, and Scott.
Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $Γ$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with $Θ\left(n^{r - r/D} \log^{r/D}(n)\right)$ edges which is universal for the family of all $r$-uniform hypergraphs with $n$ vertices and maximum degree at most $D$. This almost matches a trivial lower bound $Ω(n^{r - r/D})$ coming from the number of such hypergraphs. On a high level, we follow the strategy of Alon and Capalbo used in the graph case, that is $r = 2$. The construction of $Γ$ is deterministic and based on a bespoke product of expanders, whereas showing that $Γ$ is universal is probabilistic. Two key new ingredients are a decomposition result for hypergraphs of bounded density, based on Edmond's matroid partitioning theorem, and a tail bound for branching random walks on expanders.
A subset of $\mathbb{F}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\binom{(1 + o(1))q}{m}$ arcs of size $m \gg q^{1/2} (\log q)^{3/2}$, nearly matching a trivial lower bound $\binom{q}{m}$. This was previously known to hold for $m \gg q^{2/3} (\log q)^3$, due to Bhowmick and Roche-Newton. The lower bound on $m$ is best possible up to a logarithmic factor.
An $n$-vertex graph $G$ is locally dense if every induced subgraph of size larger than $ζn$ has density at least $d > 0$, for some parameters $ζ, d > 0$. We show that the number of induced subgraphs of $G$ with $m$ vertices and maximum degree significantly smaller than $dm$ is roughly $\binom{ζn}{m}$, for $m \ll ζn$ which is not too small. This generalises a result of Kohayakawa, Lee, Rödl, and Samotij on the number of independent sets in locally dense graphs. As an application, we slightly improve a result of Balogh, Chen, and Luo on the generalised Erdős-Rogers function for graphs with small extremal number.
Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph $H$ in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of Böttcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.
We present a short and simple proof of the celebrated hypergraph container theorem of Balogh--Morris--Samotij and Saxton--Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.
We initiate the systematic study of the following Turán-type question. Suppose $Γ$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph $G \subseteq Γ$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $Γ$ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where $H$ is a bipartite graph. We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $Γ$ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other part, all other vertices in that part have degree at most $d$, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of $Γ$. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs.
We show that if $n$ is odd and $p \ge C \log n / n$, then with high probability Hamilton cycles in $G(n,p)$ span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph $G$, that is, a graph $G$ with odd $n$ vertices and minimum degree $n/2 + C$ for sufficiently large constant $C$, span its cycle space.
A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $Ω(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction.