Abstract: A graph on n vertices is e-far from a property by inspecting the induced subgraph on a random subset of at most f(e) vertices. A property is easily testable if it is strongly testable and the function f is polynomial in 1/e, otherwise it is hard. We consider the problem of characterizing the easily testable graph properties, which is wide open, and obtain several results in its study. One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard. On the other hand, we show that being a cograph, or equivalently, induced P3-freeness where P3 is a path with 3 edges, is easily testable. This settles one of the two exceptional graphs, the other being C4 (and its complement), left open in the characterization by the first author and Shapira of graphs H for which induced H-freeness is easily testable. Our techniques yield a few additional related results, but the problem of characterizing all easily testable graph properties, or even that of formulating a plausible conjectured characterization, remains open.

Abstract: Let it(G) be the number of independent sets of size t in a graph G. Engbers and Galvin asked how large it(G) could be in graphs with minimum degree at least δ. They further conjectured that when n ≥ 2δ and t ≥ 3, it(G) is maximized by the complete bipartite graph Kδ,n−δ. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.

Abstract: We describe the C 2k+1-free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n ∉ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C 2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobas [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n 0(C 2k+1) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.

Abstract: We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least \begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*} Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromatic K4 is minimized. We show that all the extremal colourings must contain monochromatic K4 only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Abstract: A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain F1 ⊂ F2. Erdýos extended this theorem to determine the largest family without a k-chain F1 ⊂ F2 ⊂ ... ⊂ Fk. Erdýos and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds. In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman’s conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

Abstract: We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O*(m2/3n2/3 + m6/11n9/11 + m + n), where the O*(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ3 without first improving it in the plane.Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then for any ϵ > 0 the bound can be improved to \[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \] For various ranges of parameters (e.g., when m = Θ(n) and q = o(n7/9)), this bound is smaller than the lower bound Ω*(m2/3n2/3 + m + n), which holds in two dimensions.We present several extensions and applications of the new bound. (i) For the special case where all the circles have the same radius, we obtain the improved bound O(m5/11+ϵn9/11 + m2/3+ϵn1/2q1/6 + m + n).(ii) We present an improved analysis that removes the subpolynomial factors from the bound when m = O(n3/2−ϵ) for any fixed ϵ < 0.(iii) We use our results to obtain the improved bound O(m15/7) for the number of mutually similar triangles determined by any set of m points in ℝ3. Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

Abstract: Let H be a k-uniform hypergraph on n vertices where n is a sufficiently large integer not divisible by k. We prove that if the minimum (k − 1)-degree of H is at least ⌊n/k⌋, then H contains a matching with ⌊n/k⌋ edges. This confirms a conjecture of Rodl, Rucinski and Szemeredi [13], who proved that minimum (k − 1)-degree n/k + O(log n) suffices. More generally, we show that H contains a matching of size d if its minimum codegree is d < n/k, which is also best possible.

Abstract: We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

Abstract: We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemeredi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr -packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing. In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

Abstract: We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobas and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

Abstract: We show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs.

Abstract: Hajnal and Szemeredi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph with | | = ks and δ( ) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ( )= min v∈V ( )d −(v)+d +(v). Our result implies the Hajnal–Szemeredi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

Abstract: The depth of a trie has been deeply studied when the source which produces the words is a simple source (a memoryless source or a Markov chain). When a source is simple but not an unbiased memoryless source, the expectation and the variance are both of logarithmic order and their dominant terms involve characteristic objects of the source, for instance the entropy. Moreover, there is an asymptotic Gaussian law, even though the speed of convergence towards the Gaussian law has not yet been precisely estimated. The present paper describes a ‘natural’ class of general sources, which does not contain any simple source, where the depth of a random trie, built on a set of words independently drawn from the source, has the same type of probabilistic behaviour as for simple sources: the expectation and the variance are both of logarithmic order and there is an asymptotic Gaussian law. There are precise asymptotic expansions for the expectation and the variance, and the speed of convergence toward the Gaussian law is optimal. The paper first provides analytical conditions on the Dirichlet series of probabilities of a general source under which this Gaussian law can be derived: a pole-free region where the series is of polynomial growth. In a second step, the paper focuses on sources associated with dynamical systems, called dynamical sources, where the Dirichlet series of probabilities is expressed with the transfer operator of the dynamical system. Then, the paper extends results due to Dolgopyat, already generalized by Baladi and Vallee, and shows that the previous analytical conditions are fulfilled for ‘most’ dynamical sources, provided that they ‘strongly differ’ from simple sources. Finally, the present paper describes a class of sources not containing any simple source, where the trie depth has the same type of probabilistic behaviour as for simple sources, even with more precise estimates.

Abstract: We describe a general framework for realistic analysis of sorting algorithms, and we apply it to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort, BubbleSort). Usually the analysis deals with the mean number of key comparisons, but here we view keys as words produced by the same source, which are compared via their symbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n^2/4 and BubbleSort to n^2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ(n log_2 n), Θ(n^2), Θ(n^2 log n). In these three cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely entropy and coincidence) with respect to the algorithm.

Abstract: We establish a relation between two uniform models of random k-graphs (for constant k 3) on n labeled vertices: H (k) (n;m), the random k-graph with exactly m edges, and H (k) (n;d), the random d-regular k-graph. By extending to k-graphs the switching technique of McKay and Wormald, we show that, for some range of d = d(n) and a constant c > 0, if m cnd, then one can couple H (k) (n;m) and H (k) (n;d) so that the latter contains the former with probability tending to one as n!1. In view of known results on the existence of a loose Hamilton cycle in H (k) (n;m), we conclude that H (k) (n;d) contains a loose Hamilton cycle when d logn (or just d C logn, if k = 3) and d = o(n 1=2 ).

Abstract: Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

Abstract: A univariate polynomial f over a field is decomposable if it is the composition f = g ○ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f , is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n , is more challenging and our error bounds are weaker.

Abstract: Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O ( n ) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G ( n, p ) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G ( n, p ) can be decomposed into a union of n /4 + np /2 + o ( n ) cycles and edges w.h.p. This result is asymptotically tight.

Abstract: In my recent paper [1] there is a mistake in the proof of Corollary 3. The first line of the displayed equation in that proof asserts that \[\int_G|\langle u,\pi^g v\rangle_V|^2\,\rm{d} g = \int_G\langle u\otimes u,(\pi^g\otimes \pi^g)(v\otimes v)\rangle_{V\otimes V}\, \rm{d} g.\] However, since the paper uses complex-valued representations, the integrand on the right here may not retain the absolute value of that on the left. Without this equality, the proof of Corollary 3 can no longer be reduced to an application of Lemma 2. However, it can be proved directly from Schur Orthogonality along very similar lines to the proof of Lemma 2.

Abstract: Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, which settled a conjecture of Karp and Saks.

Abstract: The authors would like to rectify a mistake made in Theorem 1.1 of their article (Behrisch, Cojaa-Oghlan & Kang 2014), published in issue 23 (3). The text below explains the changes required.

Abstract: The shadow of a system of sets is all sets which can be obtained by taking a set in the original system, and removing a single element. The Kruskal-Katona theorem tells us the minimum possible size of the shadow of , if consists of m r-element sets. In this paper, we ask questions and make conjectures about the minimum possible size of a partial shadow for , which contains most sets in the shadow of . For example, if is a family of sets containing all but one set in the shadow of each set of , how large must be?

Abstract: We analyse the first-order asymptotic growth of an = ∫10∏nj=1 4 sin2(πjx) dx. The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the 'signed' number of ways in which 0 can be represented as the sum of ∈j j for ?n ≤ j ≤ n (with j ≠ 0), with ∈j ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch. Copyright © 2014 Cambridge University Press.

Abstract: Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edgeless graph, this problem is resolved by Sperner's theorem. In this paper, we focus on the case where G is the path of length n - 1, proving that the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).

Abstract: Let F = (F 1, F 2, …, F n) be a family of n sets on a ground set S, such as a family of balls in R d. For every finite measure μ on S, such that the sets of F are measurable, the classical inclusion-exclusion formula asserts that μ (F 1 ∪ F 2 ∪ …∪ F n) = ∑I:o≠⊆[n] (−1)¦I¦+1μ(∩i ∈IF i); that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in n, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families F. We provide an upper bound valid for an arbitrary F: we show that every system F of n sets with m nonempty fields in the Venn diagram admits an inclusion-exclusion formula with m O (log2 n) terms and with ±1 coefficients, and that such a formula can be computed in m O (log2 n) expected time.

Abstract: Coding in a new metric space, called the Enomoto-Katona space, has recently been considered in connection with the study of implication structures of functional dependencies and their generalizations in relational databases. The central problem is the determination of C(n,k,d), the size of an optimal code of length n, weight k, and distance d in the Enomoto-Katona space. The value of C(n,k,d) was known only for some congruence classes of n when (k,d) ∈ {(2,3),(3,5)}. In this paper, we obtain new infinite families of optimal codes in the Enomoto-Katona space and verify a conjecture of Brightwell and Katona in certain instances. In particular, C(n,k, 2k − 1) is determined for all sufficiently large n satisfying either n ≡ 1 mod k and n(n − 1) ≡ 0 mod 2k2, or n ≡ 0 mod k. We also give complete solutions for k = 2 and determine C(n,3,5) for certain congruence classes of n with finite exceptions.

Abstract: We generalize and improve recent results by Bona and Knopfmacher and by Banderier and Hitczenko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting se- quences are Riordan arrays. In this framework, we give an alternative method for com- puting asymptotics in the supercritical case of Flajolet and Sedgewick, avoiding explicit diagonal extraction. We claim that this method is more computationally efficient. j˘1 si j where each si j 2 Si . This is a straightforward generalization of the usual definition (when d ˘ 1) of restricted composition of a natural number, and is readily encoded by the matrix (si j ) once S is fixed. There are associated counting problems involving enumeration of S -compositions according to size and number of parts, and these translate as usual to probabilistic questions with respect to the uniform distribution. One particular question is: what is the probability …n that two n-compositions have the same number of parts in each component? Bona and Knopfmacher (3) studied this problem for d ˘ 2, S1 ˘ S2 and n1 ˘ n2, obtaining exact formulae for a few special choices of S1. Banderier and Hitczenko (1) generalized the problem to the case where each Si is arbitrary, and from 2 to d -tuples, still remaining in the case n ˘ n1 where all ni are equal. They obtained an explicit uni- variate generating function via the usual diagonal extraction method in some cases. They computed first-order asymptotics for the probability by means of a Gaussian local limit theorem essentially proved by Bona and Flajolet (2). This enables higher order terms to be computed in principle when combined with the diagonal extraction step, overcoming some known serious difficulties with the latter approach. 1.1. Our contribution. We generalize and improve the above results in several ways. First, we have already generalized the problem treated by previous authors by not requiring all ni to be equal. We generalize from compositions of integers to more general combinato- rial classes. Next, we derive a generating function representation for two generalizations of the sequence construction used above, namely the functional composition schema (6) and Riordan arrays. This yields a whole class of identities involving sums of squares, many of which are apparently not listed in (8). Turning to asymptotics in the Riordan array case, we show how to more efficiently compute full asymptotic expansions by adopting a mul- tivariate approach, avoiding explicit diagonal extraction and using a small fragment of the theory of asymptotic multivariate coefficient extraction developed by Pemantle and Wil- son (12). This bears out the observation of Raichev and Wilson (14) that the usual diagonal

Abstract: No presente trabalho, busco perceber como se constitui a disputatio entre os filosofos Porfirio e Jâmblico, ambos considerados pensadores neoplatonicos no livro I da obra Resposta do mestre Abamon a carta de Porfirio dirigida a Anebo e respostas as duvidas nela expressas . Mais conhecido pelo titulo atribuido por Ficino, Sobre os misterios , a obra e uma missiva imensa cuja autoria e imputada ao filosofo sirio Jâmblico de Calcis, que traz a voz argumentativa de um certo mestre Abamon. Para tanto, procuro compreender o fato de Jâmblico construir uma personagem -- o sacerdote egipcio -- para responder as interpelacoes feitas indiretamente por Porfirio, ao questionar Anebo,um suposto discipulo de Abamon ou, na verdade, Jâmblico. Alem disso, pretendo percorrer alguns argumentos do livro I de Sobre os misterios , procurando observar os primeiros passos de sua construcao argumentativa para o esclarecimento a respeito da teurgia, tendo como forma de apresentacao a estrutura em perguntas e respostas, as erotapokriseis .

Abstract: O processo de cristianizacao do Imperio Romano dependeu, em larga medida, de acoes de enfrentamento dos cristaos contra os adeptos do paganismo e do judaismo, cujos lugares e monumentos nao apenas experimentaram um processo de dessacralizacao, mas foram amiude alvo de saques e depredacoes. Nesse sentido, os ataques aos edificios greco-romanos e judaicos, tanto em termos simbolicos quanto em termos materiais, foram uma das marcas distintivas da propria cristianizacao, que nao raro comportou episodios de coercao e de violencia contra individuos e artefatos, decerto, mas tambem contra lugares e monumentos. Tendo em vista essas consideracoes, pretendemos, neste artigo, investigar a maneira pela qual o assunto e tratado por Libânio na Oratio 30 ( Pro templis ), elaborada por volta de 386. Dirigindo-se a Teodosio, o sofista o exorta a adotar uma atitude de tolerância em materia de religiao e a preservar as instituicoes pagas, em especial os templos de Antioquia, submetidos a assaltos rotineiros por parte dos monges sirios.