Abstract: For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $\alpha>0$, there exists $\epsilon>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $\left(1-\binom{k+r-2}{k-1}^{-1}+\alpha\right)n$ and a binomial random $k$-graph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-\binom{k+r-2}{k-1}^{-1}-\epsilon}$ on the same vertex set contains the $r^{\text{th}}$ power of a tight Hamilton cycle with high probability. This result for $r=1$ was first proved by McDowell and Mycroft.

Abstract: In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

Abstract: We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kuhn and Osthus. Moreover, our proof provides a randomized quasi-polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi-polynomial algorithm for finding a tight Hamilton cycle in the random $k$-uniform hypergraph $G_{n,p}^{(k)}$ for $p \ge C \log^8 n/ n$. The proofs are based on the absorbing method and follow the strategy of Kuhn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of $p$. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

Abstract: We consider instances of long-range percolation on $\mathbb Z^d$ and $\mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to\infty$. For the model on $\mathbb Z^d$ we show that, in probability as $|x|\to\infty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(\log r)^\Delta$, where $\Delta:=1/\log_2(1/\gamma)$ for $\gamma:=s/(2d)$. For the model on $\mathbb R^d$ we show that $D(0,xr)$ is, in probability as $r\to\infty$ for any nonzero $x\in\mathbb R^d$, asymptotic to $\phi(r)(\log r)^\Delta$ for $\phi$ a positive, continuous (deterministic) function obeying $\phi(r^\gamma)=\phi(r)$ for all $r>1$. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.

Abstract: 5 For graphs G and F , write G → (F )` if any coloring of the edges of G with ` colors yields 6 a monochromatic copy of the graph F . Let positive integers h and d be given. Suppose S 7 is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h 8 times (that is, by replacing the edges of S by paths of length h+ 1). We prove that there exits 9 a graph G with no more than (log s)s edges for which G→ (S)` holds, provided 10 that s ≥ s0(h, d, `), where s0(h, d, `) is some constant that depends only on h, d, and `. We 11 also extend this result to the case in which Q is a graph with maximum degree d on q vertices 12 with the property that every pair of vertices of degree greater than 2 are distance at least h+ 1 13 apart. This complements work of Pak regarding the size Ramsey number of ‘long subdivisions’ 14 of bounded degree graphs. 15

Abstract: In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simp ...

Abstract: We study near-critical behavior in the configuration model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ...

Abstract: In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → ℝ of length n contains a pattern π ∈ 𝔖k (𝔖k is the group of permutations of k elements), iff there are indices i1 f(iy) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ 𝔖k, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than ϵn places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error ϵ-tests of (ϵ−1 log n)O(k2) query complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error ϵ-test requires at least [EQUATION] queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n1−2/(k+1)).On the other hand, there always exists a non-adaptive one-sided error ϵ-test for π ∈ 𝔖k with O(ϵ−1/kn1−1/k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = (1, 3, 2), we describe an ϵ-test with (almost tight) query complexity of [EQUATION].Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ 𝔖3, tests π-freeness by making (ϵ−1 log n)O(1) queries.For all algorithms presented here, the running times are linear in their query complexity.

Abstract: The standard additive coalescent starting with n particles is a Markov process which owns several combinatorial representations, one by Pitman as a process of coalescent forests, and one by Chassaing and Louchard as the block sizes in a parking scheme. In the coalescent forest representation, edges are added successively between a random node and a random root. In this paper, we investigate an alternative construction by, instead, adding edges between roots. This construction induces exactly the same process in terms of cluster sizes, meanwhile, it allows us to make numerous new connections with other combinatorial and probabilistic models: size biased percolation, parking scheme in a tree, increasing trees, random cuts of trees. The variety of the combinatorial objects involved justifies our interest in this construction.

Abstract: We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.

Abstract: We consider the following basic geometric problem: Given epsilon in (0,1/2), a 2-dimensional figure that consists of a black object and a white background is epsilon-far from convex if it differs in at least an epsilon fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is epsilon-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity. Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is epsilon-far from convex. We show that Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(epsilon^{-4/3}) and thus beats the Omega(epsilon^{-3/2}) sample lower bound for learning convex figures under the uniform distribution from the work of Schmeltz (Data Structures and Efficient Algorithms,1992). It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

Abstract: We study the q-state ferromagnetic Potts model on the n-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) Θ(1) for β βc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0<βu<βrc that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts model on the n-vertex complete graph satisfies: (i) Θ(1) for β<βu, (ii) Θ(n1/3) for β=βu, (iii) exp⁡(nΩ(1)) for βu<β<βrc, and (iv) Θ(log⁡n) for β≥βrc. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.

Abstract: Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of Cayley's formula for the number of $n$-vertex trees. He enumerated $d$-dimensional hypertrees weighted by the squared size of their $(d-1)$-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of $d$-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of $d$-hypertrees. In addition, we study a random $1$-out model of $d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face containing it, and show it has a negligible $d$-dimensional homology.

Abstract: Cet article aborde les strategies mobilisees par des artistes œuvrant dans le secteur de la danse et du cirque pour perdurer dans une activite scenique. Si le maintien dans la carriere est complexe pour l’ensemble des artistes, il l’est d’autant plus quand l’activite expose a une usure corporelle precoce. Deux dimensions sont examinees : d’une part, la maniere dont les artistes reamenagent leur metier en vue de menager leur corps (par le biais d’une “negociation” avec leur collectif de travail) ; d’autre part, la facon dont la pluriactivite et l’auto-emploi pallient des formes de precarisation dans leur trajectoire. Les donnees etudiees illustrent aussi un processus de redefinition des manieres de se penser en tant qu’artiste alors meme que l’activite scenique n’est plus exer­cee ou plus de la meme facon. Plus generalement, ce sont les ressorts objectifs du vieillissement en emploi d’artistes choregraphiques et de cirque, et la facon dont la reference a l’engagement artistique peut perdurer au-dela d’une acti­vite scenique effective qui sont ici analyses.

Abstract: We initiate the study of the testability of properties in\emph{arbitrary planar graphs}. We prove that \emph{bipartiteness}can be tested in constant time. The previous bound for this class of graphs was $\tilde{O}(\sqrt{n})$, and the constant-time testability was only known for planar graphs with \emph{bounded degree}. Previously used transformations of unbounded-degree sparse graphs into bounded-degree sparse graphs cannot be used to reduce the problem to the testability of bounded-degree planar graphs. Our approach extends to arbitrary minor-free graphs. Our algorithm is based on random walks. The challenge here is to analyze random walks for a class of graphs that has good separators, i.e., bad expansion. Standard techniques that use a fast convergence to a uniform distribution do not work in this case. Roughly speaking, our analysis technique self-reduces the problem of finding an odd-length cycle in a autograph $G$ induced by a collection of cycles to another multigraph $G'$ induced by a set of shorter odd-length cycles, in such a way that when a random walks finds a cycle in $G'$ with probability $p >, 0$, then it does so with probability $\lambda(p)>0$ in $G$. This reduction is applied until the cycles collapse to self-loops that can be easily detected.

Abstract: We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to those random graphs, such as their number of connected components or the number of vertices of their dual complexes, as $p \to \infty$. The techniques involved in the study of these quantities also yield a Central Limit Theorem for the genus of a uniform map of order $p$, as $p \to \infty$.

Abstract: It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

Abstract: Suppose the edges of the complete graph on $n$ vertices are assigned a uniformly chosen random ordering. Let $X$ denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by Lavrov and Loh that this quantity is non-zero with probability at least $1/e-o(1)$, and conjectured that $X$ is asymptotically almost surely non-zero. In this paper, we prove their conjecture. We further prove a partial result regarding the limiting behaviour of $X$, suggesting that $X/n$ is log-normal in the limit as $n\\rightarrow\\infty$. A key idea of our proof is to show a certain relation between $X$ and its size-biased distribution. This relies heavily on estimates for the third moment of $X$.

Abstract: Biased Maker-Breaker games, introduced by Chvatal and Erdos, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container-type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H-building games studied by Bednarska and Luczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game-theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.

Abstract: Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,�), where each element x ∈ X lies in t randomly selected sets of �, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |�| ≥ |X| the hereditary discrepancy of (X,�) is with high probability O( √ tlogt); and when |X| ≫ |�| t the hereditary discrepancy of (X,�) is with high probability O(1). The

Abstract: An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $\Delta+C\Delta^{\frac{1}{2}}(\log\Delta)^4$, where $\Delta$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.

Abstract: Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper-edge sizes are bounded between $i$ and $C_{i,\epsilon} \sqrt{n}$ inclusive, then there is a list edge coloring using $(1 + \epsilon) \frac{n}{i - 1}$ colors. The dependence on $n$ in the upper bound is optimal (up to the value of $C_{i,\epsilon}$).

Abstract: Given a fixed graph H, what is the (exponentially small) probability that the number X_H of copies of H in the binomial random graph G_{n,p} is at least twice its mean? Studied intensively since the mid 1990s, this so-called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of \Pr(X_H \ge (1+\epsilon) \E X_H) for fixed \epsilon>0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.

Abstract: We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=\omega(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.

Abstract: I Introduction In the last few decades, the number of men and women aspiring to be recognized as artists and live from their art in a professional manner has significantly increased in developed societies, whether we refer to actors (Menger, 2003; Pilmis, 2013), musicians (Coulangeon, 2004), visual artists (Abbing, 2002; Karttunen, 2008; Moulin, 1983; Quemin, 2013), writers (Lahire, 2006; Sapiro, 2007), filmmakers (Alexandre, 2015), dancers (Rannou/Roharik, 2006) or circus performers (Cordie