Abstract: Answering a question raised by Dudek and Pralat, we show that if pn → ∞, w.h.p., whenever G = G(n, p) is 2-edge-coloured there is a monochromatic path of length (2/3 + o(1))n. This result is optimal in the sense that 2/3 cannot be replaced by a larger constant. As part of the proof we obtain the following result. Given a graph G on n vertices with at least edges, whenever G is 2-edge-coloured, there is a monochromatic path of length at least . This is an extension of the classical result by Gerencser and Gyarfas which says that whenever Kn is 2-coloured there is a monochromatic path of length at least 2n/3.

Abstract: We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemeredi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

Abstract: We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order n 3/4 to a constant. We show other examples for which a non-recursive, one-time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost. We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.

Abstract: This paper addresses the following question for a given graph H: what is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger’s Conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that f(Kt) = ct √ lnt. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then f(H) 6 3.895 √ lndt. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) 6 t + 6.291q (where the coefficient of 1 in thet term is best possible). 2010 Mathematics Subject Classification. 05C83, 05C35, 05D40.

Abstract: . LetAn = # red nodes at time n, andXn = sum of degrees of red nodes at time n. The process {(An, Xn)}n≥1 is Markov: An+1 = An + In+1, Xn+1 = Xn + un+1 +mIn+1, where un+1 ∼ Bin (m,xn), and P (In+1 = 1 |un+1) = pun+1. After some algebra, this implies that the fraction xn satisfies xn+1 − xn = 1 n (P (xn) + ξn+1 +Rn) , where ξn+1 is a bounded noise term, and Rn is a small error term. I.e., {xn}n≥1 is a stochastic approximation process. Thus, intuitively, {xn}n≥1 behaves approximately like the solution to a stochastic version of the ODE dz/dt = P (z); and the same holds for {an}n≥1. Intuition

Abstract: Let p 1, p 2, p 3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p 1, p 2, p 3 and the points of P is Ω(n 6/11), improving the lower bound Ω(n 0.502) of Elekes and Szabo [4] (and considerably simplifying the analysis).

Abstract: Suppose that X 1, X 2, . . . are independent identically distributed Bernoulli random variables with mean p. A Bernoulli factory for a function f takes as input X 1, X 2, . . . and outputs a random variable that is Bernoulli with mean f(p). A fast algorithm is a function that only depends on the values of X 1, . . ., XT , where T is a stopping time with small mean. When f(p) is a real analytic function the problem reduces to being able to draw from linear functions Cp for a constant C > 1. Also it is necessary that Cp ⩽ 1 − e for known e > 0. Previous methods for this problem required extensive modification of the algorithm for every value of C and e. These methods did not have explicit bounds on as a function of C and e. This paper presents the first Bernoulli factory for f(p) = Cp with bounds on as a function of the input parameters. In fact, sup p∈[0,(1−e)/C] ≤ 9.5e−1 C. In addition, this method is very simple to implement. Furthermore, a lower bound on the average running time of any Cp Bernoulli factory is shown. For e ⩽ 1/2, sup p∈[0,(1−e)/C] ≥0.004Ce−1, so the new method is optimal up to a constant in the running time.

Abstract: For positive integers $n$ and $q$ and a monotone graph property $\cA$, we consider the two player, perfect information game $\WC(n,q,\cA)$, which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of the complete graph $K_n$ which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining $q$ edges go back to Waiter. If at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\cA$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker-Chooser games) for a variety of natural graph theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter $q$ is close to $n$ and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if $q \leq (1 - \varepsilon) n$, then Client can avoid connected components of order $c \varepsilon^{-2} \ln n$ for some absolute constant $c > 0$, whereas, for $q \geq (1 + \varepsilon) n$, Waiter can force a giant, linearly sized, connected component in Client's graph. We also prove that Waiter can force Client's graph to be pancyclic for every $q \leq c n$, where $c > 0$ is an appropriate constant.

Abstract: A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

Abstract: It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$ . Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

Abstract: In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\sigma_n\,\PP(S_n\!=\!j)\!\leq\! \eta$ where $\sigma_n$ denotes the standard deviation of $S_n$ and $\eta$ is a universal constant. We compute the best possible constant $\eta\!\sim\! 0.4688$ and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for $n$ and $j$ fixed. An application to estimate the rate of convergence of Mann's fixed point iterations is presented.

Abstract: We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.

Abstract: Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan et al. have recently suggested to restrict the tester to take its queries from a smaller, typically random, subset of the inputs. This model is called active testing, in resemblance of active learning. Active testing gets more dicult as the size of the set we can query from decreases, and the extreme case is when it is exactly the number of queries we perform (so the algorithm actually has no choice). This is known as passive testing, or testing from random examples. In their paper, Balcan et al. have shown that active and passive testing of dictator functions is as hard as learning them, and requires (log n) queries (unlike the classic model, in which it can be done in a constant number of queries). We extend this result to k-linear functions, proving that passive and active testing of them requires ( k logn) queries, assuming k is not too large. Other classes of functions we consider are juntas, partially symmetric functions, linear functions, and low degree polynomials. For linear functions we provide tight bounds on the query complexity in both active and passive models (which asymptotically dier). The analysis for low degree polynomials is less complete and the exact query complexity is given only for passive testing. In both these cases, the query complexity for passive testing is essentially equivalent to that of learning. For juntas and partially symmetric functions, that is, functions that depend on a small number of variables and potentially also on the Hamming weight of the input, we provide some lower and upper bounds for the dierent models. When the functions depend on a constant number of variables, our analysis for both families is asymptotically tight. Moreover, the family of partially symmetric functions is the first example for which the query complexities in all these models are asymptotically dierent. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdfis‐Rado results on -systems.

Abstract: An old conjecture of Z. Tuza says that for any graph G , the ratio of the minimum size, τ 3 ( G ), of a set of edges meeting all triangles to the maximum size, ν 3 ( G ), of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive α there are arbitrarily large graphs G of positive density satisfying τ 3 ( G ) > (1 − o (1))| G |/2 and ν 3 ( G ) G |/4.

Abstract: The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are $$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$ where ⌊ A ⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: $$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$ where $${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$ in which the infimum is taken over cubic coverings { C i } of A ∩ C . Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min( k , D ( A )); if we assume D ( A ) = D ( A ), then the mass dimension of A under the typical orthogonal projection is equal to min( k , D ( A )). This work extends recent work of Y. Lima and C. G. Moreira.

Abstract: The discrete Green's function (without boundary) is a pseudo-inverse of the combinatorial Laplace operator of a graph G = (V, E). We reveal the intimate connection between Green's function and the theory of exact stopping rules for random walks on graphs. We give an elementary formula for Green's function in terms of state-to-state hitting times of the underlying graph. Namely, where π i is the stationary distribution at vertex i, H(i, j) is the expected hitting time for a random walk starting from vertex i to first reach vertex j, and H(π, j) = ∑ k∈V π k H(k, j). This formula also holds for the digraph Laplace operator. The most important characteristics of a stopping rule are its exit frequencies, which are the expected number of exits of a given vertex before the rule halts the walk. We show that Green's function is, in fact, a matrix of exit frequencies plus a rank one matrix. In the undirected case, we derive spectral formulas for Green's function and for some mixing measures arising from stopping rules. Finally, we further explore the exit frequency matrix point of view, and discuss a natural generalization of Green's function for any distribution τ defined on the vertex set of the graph.

Abstract: Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the inmum of all non-negative reals c such that the subfamily of F comprising hy- pergraphs H with minimum degree at least c jV (H)j r−1 � has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs. Luczak and Thomasse recently proved that the chromatic threshold of near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turan number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few innite families of nondegenerate hypergraphs whose Turan number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing fabc,abd,cdeg, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property ofber bundles which is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a generalized Kneser hypergraph. Using methods from extremal set theory, we prove that these generalized Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemeredi for graphs and might be of independent interest. Many open problems remain.

Abstract: A k-uniform hypergraph H = (V, E) is called l-orientable if there is an assignment of each edge e is an element of E to one of its vertices v is an element of e such that no vertex is assigned more than l edges. Let H-n,H-m,H-k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the l-orientability of H-n,H-m,H-k for all k >= 3 and l >= 2, that is, we determine a critical quantity c(*)k,l such that with probability 1-o(1) the graph H-n,H-cn,(k) has an l-orientation if c c(k,l)(*) . Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

Abstract: A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).

Abstract: We consider connectivity properties and asymptotic slopes for certain random directed graphs on Z in which the set of points Co that the origin connects to is always infinite. We obtain conditions under which the complement of Co has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in 2 dimensions.

Abstract: Let G 1 × G 2 denote the strong product of graphs G 1 and G 2, that is, the graph on V(G 1) × V(G 2) in which (u 1, u 2) and (v 1, v 2) are adjacent if for each i = 1, 2 we have ui = vi or u i v i ∈ E(G i ). The Shannon capacity of G is c(G) = limn → ∞ α(Gn )1/n , where Gn denotes the n-fold strong power of G, and α(H) denotes the independence number of a graph H. The normalized Shannon capacity of G is Alon [1] asked whether for every e 1 − e. We show that the answer is no.

Abstract: We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and col ...

Abstract: The authors would like to rectify a mistake made on the second page, line 50 of their article. The text below explains the changes required. The constant 0.628 on line 50 of page 2 should be 0.638.