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  • 1.
    Behrstock, Jason
    et al.
    Lehman College, CUNY.
    Falgas-Ravry, Victor
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hagen, Mark
    University of Cambridge.
    Susse, Tim
    University of Nebraska-Lincoln.
    Global Structural Properties of Random Graphs2018In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 5, p. 1411-1441Article in journal (Refereed)
    Abstract [en]

    We study two global structural properties of a graph , denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erd ˝os–Rényi random graph model G(n, p), proving the existence of a sharp threshold for a random graph to have the AS property asymptotically almost surely, and giving fairly tight bounds for the corresponding threshold for the CFS property. As an application of our results, we show that for any constant p and any ∈ G(n, p), the right-angled Coxeter group W asymptotically almost surely has quadratic divergence and thickness of order 1, generalizing and strengthening a result of Behrstock–Hagen–Sisto [8]. Indeed, we show that at a large range of densities a random right-angled Coxeter group has quadratic divergence. 1

  • 2. Björklund, Johan
    et al.
    Falgas-Ravry, Victor
    Holmgren, Cecilia
    On percolation in one-dimensional stable Poisson graphs2015In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 20, no 50, p. 1-6Article in journal (Refereed)
    Abstract [en]

    Equip each point x of a homogeneous Poisson point process P on R with D-x edge stubs, where the D-x are i.i.d. positive integer-valued random variables with distribution given by mu. Following the stable multi-matching scheme introduced by Deijfen, Haggstrom and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph G on the vertex set P. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Haggstrom and Holroyd [1] on percolation (the existence of an infinite connected component) in G. We prove that percolation may occur a.s. even if mu has support over odd integers. Furthermore, we show that for any epsilon > 0, there exists a distribution mu such that mu ({1}) > 1 - epsilon, but percolation still occurs a.s..

  • 3.
    Falgas-Ravry, Victor
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold2013In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 18, no paper 83, p. 1-22Article in journal (Refereed)
    Abstract [en]

    Let S-n,S-k denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k = k (n) points of the process nearest to it. In this paper we show that if P (S-n,S-k connected) > n(-gamma 1) then the probability that S-n,S-k contains a pair of 'small' components 'close' to each other is o(n (c1)) (in a precise sense of 'small' and 'close'), for some absolute constants gamma(1) > 0 and c(1) > 0. This answers a question of Walters [13]. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of S-n,S-k below the connectivity threshold is asymptotically Poisson.

  • 4.
    Falgas-Ravry, Victor
    School of Mathematical Sciences Queen Mary University of London, London E1 4NS, UK.
    Minimal weight in union-closed families2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 18, no 1, p. P95-Article in journal (Refereed)
    Abstract [en]

    Let Omega be a finite set and let S subset of P(Omega) be a set system on Omega. For x is an element of Omega, we denote by d(S)(x) the number of members of S containing x.Along-standing conjecture of Frankl states that if S is union-closed then there is some x is an element of Omega with d(S)(x)>= 1/2|S|. We consider a related question. Define the weight of a family S to be w(S) := A.S|A|.SupposeSisunion-closed. How small can w(S) be? Reimer showed w(S) >= 1/2|S|log(2)|S|, and that this inequality is tight. In this paper we show how Reimer's bound may be improved if we have some additional information about the domain Omega of S: if S separates the points of its domain, then w(S) >= ((vertical bar Omega vertical bar)(2)). This is stronger than Reimer's Theorem when |Omega| > root|S|log(2)|S|. In addition we constructa family of examples showing the combined bound on w(S)istightexcept in the region |Omega| = Theta(root|S|log(2)|S|), where it may be off by a multiplicative factor of 2. Our proof also gives a lower bound on the average degree: if S is a point-separating union-closed family on Omega, then 1/ |Omega|Sigma(x is an element of Omega)d(S)(x)>= 1/2 root|S|log(2)|S| broken vertical bar O(1), and this is best possible except for a multiplicative factor of 2.

  • 5.
    Falgas-Ravry, Victor
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the codegree density of complete 3-graphs and related problems2013In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 20, no 4, p. Article number: P28-Article in journal (Refereed)
    Abstract [en]

    Given a 3-graph F, its codegree threshold co-ex(n, F) is the largest number d - d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d triples but which contains no member of F as a subgraph. The limit [GRAPHICS] is known to exist and is called the codegree density of F. In this paper we generalise a construction of Czygrinow and Nagle to bound below the codegree density of complete 3-graphs: for all integers s >= 4, the codegree density of the complete 3-graph on s vertices K-s satisfies [GRAPHICS] We then provide constructions based on Steiner triple systems which show that if this lower bound is sharp, then we do not have stability in general. In addition we prove bounds on the codegree density for two other in finite families of 3-graphs.

  • 6.
    Falgas-Ravry, Victor
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sperner's Problem for G-Independent Families2015In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 24, no 3, p. 528-550Article in journal (Refereed)
    Abstract [en]

    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).

  • 7. Falgas-Ravry, Victor
    et al.
    Kittipassorn, Teeradej
    Korándi, Dániel
    Letzter, Shoham
    Narayanan, Bhargav P
    Separating path systems2014In: Journal of Combinatorics, ISSN 2156-3527, E-ISSN 2150-959X, Vol. 5, no 3, p. 335-354Article in journal (Refereed)
    Abstract [en]

    We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every nn-vertex graph admits a separating path system of size linear in nn and we prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.

  • 8.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Lo, Allan
    Subgraphs with large minimum ℓ-degree in hypergraphs where almost all ℓ-degrees are large2018In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, no 2, article id P2.18Article in journal (Refereed)
    Abstract [en]

    Let G be an r-uniform hypergraph on n vertices such that all but at most ε(n ℓ) ℓ-subsets of vertices have degree at least p(n-ℓ r-ℓ). We show that G contains a large subgraph with high minimum ℓ-degree.

  • 9.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Vanderbilt Univ, Dept Math, Nashville, TN 37240 USA.
    Marchant, Edward
    Pikhurko, Oleg
    Vaughan, Emil R.
    The Codegree Threshold for 3-Graphs with Independent Neighborhoods2015In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 29, no 3, p. 1504-1539Article in journal (Refereed)
    Abstract [en]

    Given a family of 3-graphs F, we define its codegree threshold coex(n, F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F-3,F-2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe, and cde. In this paper, we give two proofs that coex(n, {F-3,F-2}) = - (1/3 + o(1))n, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large n: coex(n, {F-3,F-2}) = [n/3] - 1 if n is congruent to 1 modulo 3, and [n/3] otherwise. In addition we determine the set of codegree-extremal configurations for all sufficiently large n.

  • 10.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Random subcube intersection graphs I: cliques and covering2016In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 23, no 3, article id P3.43Article in journal (Refereed)
    Abstract [en]

    We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube Qd to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model 'random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube Qd and for the appearance of s-cliques. In addition we pose a number of open problems.

  • 11.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Markström, Klas
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Verstraëte, Jacques
    University of California-San Diego.
    Full subgraphs2018In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 88, no 3, p. 411-427Article in journal (Refereed)
    The full text will be freely available from 2020-12-31 17:27
  • 12.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    O'Connell, Kelly
    Uzzell, Andrew
    Multicolor containers, extremal entropy, and counting2019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, no 4, p. 676-720Article in journal (Refereed)
    Abstract [en]

    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 simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colorings of very general objects, which include both vertex- and edge-colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.

  • 13.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pikhurko, Oleg
    University of Warwick.
    Vaughan, Emil
    Center for Discrete Mathematics, University of London.
    Volec, Jan
    McGill University.
    The codegree threshold of K_4^-2017In: The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17) / [ed] Drmota Michael; Kang Mihyun; Krattenthaler Christian; Nešetřil Jaroslav, Elsevier, 2017, Vol. 61, p. 407-413Conference paper (Refereed)
    Abstract [en]

    The codegree threshold ex2(n, F) of a non-empty 3-graph F is the minimum d = d(n) such that every 3-graph on n vertices in which every pair of vertices is contained in at least d+ 1 edges contains a copy of F as a subgraph. We study ex2(n, F) when F = K − 4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that

    ex2(n, K− 4 ) = n 4 + o(n).

    This settles in the affirmative a conjecture of Nagle [20]. In addition, we obtain a stability result: for every near-extremal configurations G, there is a quasirandom tournament T on the same vertex set such that G is close in the edit distance to the 3-graph C(T) whose edges are the cyclically oriented triangles from T. For infinitely many values of n, we are further able to determine ex2(n, K− 4 ) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.

  • 14.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vaughan, Emil R.
    Univ London, Sch Elect Engn & Comp Sci, London E1 4NS, England.
    Applications of the semi-definite method to the Turan density problem for 3-graphs2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 1, p. 21-54Article in journal (Refereed)
    Abstract [en]

    In this paper, we prove several new Turan density results for 3-graphs with independent neighbourhoods. We show: pi(K-4, C-5, F-3,F-2) = 12/49, pi(K-4, F-3,F-2) = 5/18 and pi(J(4), F-3,F-2) = pi(J(5), F-3,F-2) = 3/8, where J(t) is the 3-graph consisting of a single vertex x together with a disjoint set A of size t and all (vertical bar A vertical bar 2) 3-edges containing x. We also prove two Turan density results where we forbid certain induced subgraphs: pi(F-3,F-2, induced K-4(-)) = 3/8 and pi(K-5, 5-set spanning exactly 8 edges) = 3/4. The latter result is an analogue for K-5 of Razborov's result that pi(K-4, 4-set spanning exactly 1 edge) = 5/9. We give several new constructions, conjectures and bounds for Turan densities of 3-graphs which should be of interest to researchers in the area. Our main tool is 'Flagmatic', an implementation of Razborov's semi-definite method, which we are making publicly available. In a bid to make the power of Razborov's method more widely accessible, we have tried to make Flagmatic as user-friendly as possible, hoping to remove thereby the major hurdle that needs to be cleared before using the semi-definite method. Finally, we spend some time reflecting on the limitations of our approach, and in particular on which problems we may be unable to solve. Our discussion of the 'complexity barrier' for the semi-definite method may be of general interest.

  • 15.
    Falgas-Ravry, Victor
    et al.
    Queen Mary University of London, School of Mathematical Sciences.
    Walters, Mark
    Queen Mary University of London, School of Mathematical Sciences.
    Sharpness in the k-nearest-neighbours random geometric graph model2012In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 44, no 3, p. 617-634Article in journal (Refereed)
    Abstract [en]

    Let Sn,k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn,k is connected is at least ε, then the probability that Sn,k+Cis connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequences(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn is s-connected (i.e. remains connected after the deletion of any s - 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

  • 16.
    Falgas–Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhao, Yi
    Codegree thresholds for covering 3-uniform hypergraphs2016In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 30, no 4, p. 1899-1917Article in journal (Refereed)
    Abstract [en]

    Given two 3-uniform hypergraphs F and G = (V, E), we say that G has an F-covering if we can cover V with copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V is contained in at least d triples from E. Define c(2)(n, F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. Determining c(2)(n, F) is a natural problem intermediate (but distinct) from the well-studied Turan problems and tiling problems. In this paper, we determine c(2)(n, K-4) (for n > 98) and the associated extremal configurations (for n > 998), where K-4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c(2)(n, F) which are apart by at most 2 in the cases where F is K-4(-) (K-4 with one edge removed), K-5(-), and the tight cycle C-5 on 5 vertices.

  • 17.
    Ravry, Victor Falgas
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vaughan, Emil R.
    Turan H-densities for 3-graphs2012In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, no 3, p. P40-Article in journal (Refereed)
    Abstract [en]

    Given an r-graph H on h vertices, and a family F of forbidden subgraphs, we define ex H (n, F) to be the maximum number of induced copies of H in an F-free r-graph on n vertices. Then the Turan H-density of F is the limit pi(H)(F) = (lim)(n ->infinity) ex(H)(n, F)/((n)(h)) This generalises the notions of Turan-density (when H is an r-edge), and inducibility (when F is empty). Although problems of this kind have received some attention, very few results are known. We use Razborov's semi-definite method to investigate Turan H-densities for 3-graphs. In particular, we show that pi(-)(K4)(K-4) = 16/27, with Turans construction being optimal. We prove a result in a similar flavour for K-5 and make a general conjecture on the value of pi(Kt)-(K-t). We also establish that pi(4.2)(empty set) = 3/4, where 4: 2 denotes the 3-graph on 4 vertices with exactly 2 edges. The lower bound in this case comes from a random geometric construction strikingly different from previous known extremal examples in 3-graph theory. We give a number of other results and conjectures for 3-graphs, and in addition consider the inducibility of certain directed graphs. Let (S) over right arrow (k) be the out-star on k vertices; i.e. the star on k vertices with all k 1 edges oriented away from the centre. We show that pi((S) over right arrow3)(empty set) = 2 root 3 - 3, with an iterated blow-up construction being extremal. This is related to a conjecture of Mubayi and Rodl on the Turan density of the 3-graph C-5. We also determine pi((S) over right arrowk) (empty set) when k = 4, 5, and conjecture its value for general k.

1 - 17 of 17
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