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
Given a graph (Formula presented.), its auxiliary square-graph (Formula presented.) is the graph whose vertices are the non-edges of (Formula presented.) and whose edges are the pairs of non-edges which induce a square (i.e., a 4-cycle) in (Formula presented.). We determine the threshold edge-probability (Formula presented.) at which the Erdős–Rényi random graph (Formula presented.) begins to asymptotically almost surely (a.a.s.) have a square-graph with a connected component whose squares together cover all the vertices of (Formula presented.). We show (Formula presented.), a polylogarithmic improvement on earlier bounds on (Formula presented.) due to Hagen and the authors. As a corollary, we determine the threshold (Formula presented.) at which the random right-angled Coxeter group (Formula presented.) a.a.s. becomes strongly algebraically thick of order 1 and has quadratic divergence.
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..
Motivated by problems in percolation theory, we study the following two-player positional game. Let ?(mxn) be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board ?(mxn), while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on ?(mxn). Given m, n is an element of N, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on ?(mxn)? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If >= 2, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2, )-crossing game on ?x(+1 for any is an element of N. pqqqmq)mIf p <= 2q - 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on ?mxn for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.
Let Lambda be an infinite connected graph, and let v(0) be a vertex of Lambda. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of Lambda are marked as unsafe. On each of her turns, Maker marks p unsafe edges as safe, while on each of his turns Breaker takes q unsafe edges and deletes them from the graph. Breaker wins if at any time in the game the component containing v(0) becomes finite. Otherwise if Maker is able to ensure that v(0) remains in an infinite component indefinitely, then we say she has a winning strategy. This game can be thought of as a variant of the celebrated Shannon switching game. Given (p, q) and (Lambda, v(0)), we would like to know: which of the two players has a winning strategy?
Our main result in this paper establishes that when Lambda = Z(2) and v(0) is any vertex, Maker has a winning strategy whenever p >= 2q, while Breaker has a winning strategy whenever 2p <= q. In addition, we completely determine which of the two players has a winning strategy for every pair (p, q) when Lambda is an infinite d -regular tree. Finally, we give some results for general graphs and lattices and pose some open problems. (C) 2020 Elsevier Inc. All rights reserved.
A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and Bollobás asked for the value of the least p⋆ such that for all p > p⋆ and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p⋆. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f1, G(p), the infimum over all of the probability that is connected. We determine f1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.
An (n,s,q)-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n,s,q)-graphs. We give a general lower bound construction for this problem for many pairs (s,q), which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the (s,q)=(4,6a+3) case of the problem (for a≥2); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for ‘sparse’ multigraphs (i.e. when q≤2(s2)). Finally we introduce some tools that are likely to be useful for attacking the problem in general.
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.
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.
A multigraph G is an (s,q)-graph if every s-set of vertices in G supports at most q edges of G, counting multiplicities. Mubayi and Terry posed the problem of determining the maximum of the product of the edge-multiplicities in an (s,q)-graph on n vertices. We give an asymptotic solution to this problem for the family (s,q)=(2r,a(2r2)+ex(2r,Kr+1)−1) with r,a ∈ Z≥2. This greatly generalises previous results on the problem due to Mubayi and Terry and to Day, Treglown and the author, who between them had resolved the special case r=2. Our result asymptotically confirms an infinite family of cases in (and overcomes a major obstacle to a resolution of) a conjecture of Day, Treglown and the author.
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.
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).
We use the hypergraph container theory of Balogh–Morris–Samotij and Saxton–Thomason to obtain general rectilinear approximations and volume estimates for sequences of bodies closed under certain families of projections. We give a number of applications of our results, including a multicolour generalisation of a theorem of Hatami, Janson and Szegedy on the entropy of graph limits. Finally, we raise a number of questions on geometric and analytic approaches to containers.
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.
Let V be an n-set, and let X be a random variable taking values in the powerset of V. Suppose we are given a sequence of random coupons X1,X2,…, where the Xi are independent random variables with distribution given by X. The covering time T is the smallest integer t≥0 such that ⋃ti=1Xi=V. The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.
In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.
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.
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.
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.
Let denote the power set of [n], ordered by inclusion, and let denote the random poset obtained from by retaining each element from independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in . Finally, we obtain a number of results on the Ramsey properties of the random poset .
We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c(1)(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree delta(1)(G) > d then every vertex of G is contained in a copy of F in G?
We asymptotically determine c(1)(n, F) when F is the generalized triangle K-4((3)), and we give close to optimal bounds in the case where F is the tetrahedron K-4((3)) (the complete 3-graph on 4 vertices).
This latter problem turns out to be a special instance of the following problem for graphs: Given an nvertex graph G with m> n(2)/4 edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.
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.
A random graph model on a host graph (Formula presented.) is said to be 1-independent if for every pair of vertex-disjoint subsets (Formula presented.) of (Formula presented.), the state of edges (absent or present) in (Formula presented.) is independent of the state of edges in (Formula presented.). For an infinite connected graph (Formula presented.), the 1-independent critical percolation probability (Formula presented.) is the infimum of the (Formula presented.) such that every 1-independent random graph model on (Formula presented.) in which each edge is present with probability at least (Formula presented.) almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that (Formula presented.) tends to a limit in (Formula presented.) as (Formula presented.), and they asked for the value of this limit. We make progress on a related problem by showing that (Formula presented.) In fact, we show that the equality above remains true if the sequence of complete graphs (Formula presented.) is replaced by a sequence of weakly pseudorandom graphs on (Formula presented.) vertices with average degree (Formula presented.). We conjecture the answer to Balister and Bollobás's question is also (Formula presented.).
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.
The codegree threshold ex2 (n, F) of a 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 if n is sufficiently large, then
ex2 (n, K-4)⩽ (n + 1)/4.
This settles in the affirmative a conjecture of Nagle [Congressus Numerantium, 1999, pp. 119-128]. In addition, we obtain a stability result: for every near-extremal configuration G, there is a quasirandom tournament T on the same vertex set such that G is o(n3)-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.
Following Bradonji´c and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is a log n for some fixed a>1. Each vertex is added with probability p to a set A0 of initially infected vertices. Vertices subsequently become infected if they have at least θa log n infected neighbours. Here p, θ ∈ [0, 1] are taken to be fixed constants.
We show that if θ <(1+p)/2, then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for θ >(1+p)/2, even if one adversarially infects every vertex inside a ball of radius O(√log n), with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.
In addition we give some bounds on the (a, p, θ) regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.
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.
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).
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.
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.