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Abstract: The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including the Lovász number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points). We show that every spectral point admits a probabilistic refinement and characterize the functions arising in this way. This reveals that the asymptotic spectrum can be parameterized with a convex set and the evaluation function at every graph is logarithmically convex. One consequence is that for any incomparable pair of spectral points f and g there exists a third one h and a graph G such that h(G) < min{f(G),g(G)}, thus h gives a better upper bound on the Shannon capacity of G. In addition, we show that the (logarithmic) probabilistic refinement of a spectral point on a fixed graph is the entropy function associated with a convex corner. PubDate: 2021-08-31
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Abstract: Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This is a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved that every graph with no odd Kt minor is \(O(t\sqrt {\log t} )\) -colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no Kt minor, we make the first improvement on this bound by showing that every graph with no odd Kt minor is O(t(logt)β)-colorable for every β > 1/4. PubDate: 2021-08-31
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Abstract: We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line systems in 19 and 20 dimensions to 74 and 94, respectively. PubDate: 2021-08-31
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Abstract: The main result of this paper is that, if Γ is a finite connected 4-valent vertex- and edge-transitive graph, then either Γ is part of a well-understood family of graphs, or every non-identity automorphism of Γ fixes at most 1/3 of the vertices. As a corollary, we get a similar result for 3-valent vertex-transitive graphs. PubDate: 2021-08-31
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Abstract: Pokrovskiy conjectured that there is a function f: ℕ → ℕ such that any 2k-strongly-connected tournament with minimum out and in-degree at least f(k) is k-linked. In this paper, we show that any (2k + 1)-strongly-connected tournament with minimum out-degree at least some polynomial in k is k-linked, thus resolving the conjecture up to the additive factor of 1 in the connectivity bound, but without the extra assumption that the minimum in-degree is large. Moreover, we show the condition on high minimum out-degree is necessary by constructing arbitrarily large tournaments that are (2.5k − 1)-strongly-connected tournaments but are not k-linked. PubDate: 2021-08-31
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Abstract: Besides various asymptotic results on the concept of sum-product bases in the set of non-negative integers ℕ, we investigate by probabilistic arguments the existence of thin sets A, A′ of non-negative integers such that AA + A = ℕ and A′A′ + A′A′ = ℕ. PubDate: 2021-08-31
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Abstract: Let f be a smooth real function with strictly monotone first k derivatives. We show that for a finite set A, with ∣A + A∣ ≤K∣A∣, $$\left {{2^k}f(A) - ({2^k} - 1)f(A)} \right { \gg _k}\,{\left A \right ^{k + 1 - o(1)}}/{K^{{O_k}(1)}}.$$ We deduce several new sum-product type implications, e.g. that A+A being small implies unbounded growth for a many enough times iterated product set A ⋯ A. PubDate: 2021-08-31
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Abstract: We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999). PubDate: 2021-08-31
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Abstract: Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I∣G∣, where H* denotes the complex conjugate transpose of H, and I∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family of BH(G, h) matrices in which G does not need to be abelian. The second and the third constructions are two families of BH(G, h) matrices in which G is a finite local ring. PubDate: 2021-08-31
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Abstract: Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime ) = O({n^{3/2 - {1 \over {2s}}}})\) . This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s,k ≥ 1, we show that \({\rm{ex}}(n,L) = \Theta ({n^{1 + {s \over <Stack><Subscript>+ 1</Subscript></Stack>}}})\) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(nr) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing Hk for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, Hk−1) = O(n1+1/k−δ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C4 as a subgraph satisfies ex(n, H) = o(n2−1/r). PubDate: 2021-08-26
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Abstract: Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing \(K_{s,t}^\prime \) for the subdivision of the bipartite graph Ks,t, we show that \({\rm{ex}}(n,K_{s,t}^\prime) = O({n^{3/2 - {1 \over {2s}}}})\) . This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that \({\rm{ex}}(n,L) = \Theta ({n^{1 + {s \over {sk + 1}}}})\) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(nr) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing Hk for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, Hk−1) = O(n1+1/k−δ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C4 as a subgraph satisfies ex(n, H) = o(n2−1/r). PubDate: 2021-08-13
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Abstract: We present the solution of a long-standing open question by giving an explicit construction of an infinite family of \(\mathbb{M}\) -vertex cubic graphs that have diameter \(\left[ {1 + o\left( 1 \right)} \right]{\log _2}\mathbb{M}\) . Then, for every K in the form K = ps + 1, where p can be any prime [including 2] and s any positive integer, we extend the techniques to construct an infinite family of K-regular graphs on \(\mathbb{M}\) vertices with diameter \(\left[ {1 + o\left( 1 \right)} \right]{\log _{K - 1}\mathbb{M}}\) . PubDate: 2021-06-01
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Abstract: We prove that for every integer t ⩾ 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius. PubDate: 2021-06-01
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Abstract: How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G' These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph H if it is known that any collection of a few edges of H has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Cirão, Letzter and Sahasrabudhe. PubDate: 2021-06-01
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Abstract: In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ωω. Big Ramsey degrees of finite chains in all other countable ordinals are infinite. PubDate: 2021-06-01
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Abstract: We prove for every graph H there exists ɛ > 0 such that, for every graph G with G ≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ G neighbours, or there are two disjoint sets A, B ⊆ V(G) with A , B ≥ɛ G such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least G c. This is related to the Erdős-Hajnal conjecture. PubDate: 2021-06-01
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Abstract: We prove a bijection between the triangulations of the 3-dimensional cyclic poly-tope C(n + 2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We also give an algorithm to efficiently enumerate all persistent graphs on n vertices and thus all triangulations of C(n + 2, 3). PubDate: 2021-06-01
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Abstract: Robertson and Seymour proved that the family of all graphs containing a fixed graph H as a minor has the Erdős-Pósa property if and only if H is planar. We show that this is no longer true for the edge version of the Erdős-Pósa property, and indeed even fails when H is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos. PubDate: 2021-04-01
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Abstract: A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the ‘correct number’ of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts, the global one and the local one. We first observe that if T has the correct global count of H and h ≥ 7 then quasirandomness of T is only forced if H is transitive. The next natural question when studying quasirandom objects asks whether possessing the correct local counts of H is enough to force quasirandomness of T. A tournament H is said to be locally forcing if it has this property. Variants of the local forcing problem have been studied before in both the graph and hypergraph settings. Perhaps the closest analogue of our problem was considered by Simonovits and Sós who looked at whether having ‘correct counts’ of a fixed graph H as an induced subgraph of G implies G must be quasirandom, in an appropriate sense. They proved that this is indeed the case when H is regular and conjectured that it holds for all H (except the path on 3 vertices). Contrary to the Simonovits-Sós conjecture, in the tournament setting we prove that a constant proportion of all tournaments are not locally forcing. In fact, any locally forcing tournament must itself be strongly quasirandom. On the other hand, unlike the global forcing case, we construct infinite families of non-transitive locally forcing tournaments. PubDate: 2021-04-01
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Abstract: We prove a quantitative, finitary version of Trofimov’s result that a connected, locally finite vertex-transitive graph Γ of polynomial growth admits a quotient with finite fibres on which the action of Aut(Γ) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph Γ of large diameter admits a quotient with fibres of small diameter on which the action of Aut(Γ) is virtually abelian with vertex stabilisers of bounded size. We also show that Γ has moderate growth in the sense of Diaconis and Saloff-Coste, which is known to imply that the mixing and relaxation times of the lazy random walk on Γ are quadratic in the diameter. These results extend results of Breuillard and the second author for finite Cayley graphs of large diameter. Finally, given a connected, locally finite vertex-transitive graph Γ exhibiting polynomial growth at a single, sufficiently large scale, we describe its growth at subsequent scales, extending a result of Tao and an earlier result of our own for Cayley graphs. In forthcoming work we will give further applications. PubDate: 2021-04-01
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