Authors:Jacob Fox; János Pach; Andrew Suk Abstract: Abstract We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3637-x

Authors:Morgan Rodgers; Leo Storme; Andries Vansweevelt Abstract: Abstract We look at a generalization of Cameron-Liebler line classes to sets of k-spaces, focusing on results in PG(2k+1, q). Here we obtain a connection to k-spreads which parallels the situation for line classes in PG(3,q). After looking at some characterizations of these sets and some of the difficulties that arise in contrast to the known results for line classes, we give some connections to various other geometric objects including k-spreads and Erdős–Ko–Rado sets, and prove results concerning the existence of these objects. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3482-y

Authors:Terence Tao; Van Vu Abstract: Abstract Let M n =(ξ ij )1≤i,j≤n be a real symmetric random matrix in which the upper-triangular entries ξ ij , i < j and diagonal entries ξ ii are independent. We show that with probability tending to 1, M n has no repeated eigenvalues. As a corollary, we deduce that the Erdős-Rényi random graph has simple spectrum asymptotically almost surely, answering a question of Babai. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3363-4

Authors:Shaofei Du; Wenqin Xu; Guiying Yan Abstract: Abstract This paper contributes to the classification of finite 2-arc-transitive graphs. In [12], all the regular covers of complete bipartite graphs K n,n were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of K n,n , whose covering transformation group is elementary abelian group of order p 2 and whose fibre-preserving automorphism group acts 2-arc-transitively. As a result, two new infinite families of 2-arc-transitive graphs are found. Moveover, it will be explained that it seems to be infeasible to classify all such covers when the covering transformation group is an elementary abelian group of order p k for an arbitrary integer k. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3511-x

Authors:Daniel Pellicer Abstract: From a given abstract n-polytope P and a given integer k we derive two abstract polytopes Cl k (P) and \({\widetilde {Cl}_k}\left( P \right)\) of ranks n and n−1, respectively. These constructions generalise the truncation of convex polyhedra and the dual of a geometric construction yielding Petrie’s polyhedron {4,6 4}. We determine sufficient and necessary conditions to guarantee that Cl k (P) and \({\widetilde {Cl}_k}\left( P \right)\) are regular. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3518-3

Authors:Guus Regts Abstract: Abstract Based on a technique of Barvinok [4,5,6] and Barvinok and Soberón [8,9] we identify a class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs. Subsequently we give a quasi-polynomial time approximation scheme for computing these partition functions. As another application we show that the normalised partition functions of these models are continuous with respect to the Benjamini-Schramm topology on bounded degree graphs. We moreover give quasi-polynomial time approximation schemes for evaluating a large class of graph polynomials, including the Tutte polynomial, on bounded degree graphs. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3506-7

Authors:Anton Malyshev; Igor Pak Abstract: Abstract It is known that random 2-lifts of graphs give rise to expander graphs. We present a new conjectured derandomization of this construction based on certain Mealy automata. We verify that these graphs have polylogarithmic diameter, and present a class of automata for which the same is true. However, we also show that some automata in this class do not give rise to expander graphs. PubDate: 2017-03-24 DOI: 10.1007/s00493-016-3306-0

Authors:Adam Sheffer; Endre Szabó; Joshua Zahl Abstract: Abstract We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in R2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is \(O\left( {{m^{\frac{k}{{2k - 1}}}}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\) . We establish the slightly weaker bound \({O_\varepsilon }\left( {{m^{\frac{k}{{2k - 1}} + \varepsilon }}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\) on the number of incidences between m points and n (complex) algebraic curves in C2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over C. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3441-7

Authors:Eli Berger Abstract: Abstract We prove that in any graph containing no subdivision of an infinite clique there exists a partition of the vertices into two parts, satisfying the condition that every vertex has at least as many neighbors in the part not containing it as it has in its own part. PubDate: 2017-02-13 DOI: 10.1007/s00493-015-3261-1

Authors:Aart Blokhuis; Hao Chen Abstract: Abstract A set U of unit vectors is selectively balancing if one can find two disjoint subsets U + and U -, not both empty, such that the Euclidean distance between the sum of U + and the sum of U - is smaller than 1. We prove that the minimum number of unit vectors that guarantee a selectively balancing set in R n is asymptotically 1/2nlogn. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3635-z

Authors:Brandon Hanson Abstract: Abstract Guth and Katz proved that any point set P in the plane determines Ω( P /log P ) distinct distances. We show that when near to this lower bound, a point set P of the form A × A must satisfy A-A ≪ A-A 2-1/8. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3665-6

Authors:Reinhard Diestel; Malte Müller Abstract: Abstract The connected tree-width of a graph is the minimum width of a tree-decomposition whose parts induce connected subgraphs. Long cycles are examples of graphs that have small tree-width but large connected tree-width. We show that a graph has small connected tree-width if and only if it has small tree-width and contains no long geodesic cycle. We further prove a connected analogue of the duality theorem for tree-width: a finite graph has small connected tree-width if and only if it has no bramble whose connected covers are all large. Both these results are qualitative: the bounds are good but not tight. We show that graphs of connected tree-width k are k-hyperbolic, which is tight, and that graphs of tree-width k whose geodesic cycles all have length at most l are ⌊3/2l(k-1)⌋-hyperbolic. The existence of such a function h(k, l) had been conjectured by Sullivan. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3516-5

Authors:Javier Cilleruelo; Rafael Tesoro Abstract: Abstract We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3444-4

Authors:Daniele Bartoli; Massimo Giulietti; Giuseppe Marino; Olga Polverino Abstract: Abstract Explicit constructions of inFInite families of scattered F q -linear sets in PG(r-1, q t ) of maximal rank rt/2, for t ≥ 4 even, are provided. When q = 2, these linear sets correspond to complete caps in AG(r,2 t ) FIxed by a translation group of size 2rt/2. The doubling construction applied to such caps gives complete caps in AG(r+1, 2 t ) of size 2rt/2+1. For Galois spaces of even dimension greater than 2 and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a complete cap is substantially sharp. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3531-6

Authors:João Araújo; Wolfram Bentz; Edward Dobson; Janusz Konieczny; Joy Morris Abstract: Abstract We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a “normal” circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3403-0

Authors:Tristram C. Bogart; Edward D. Kim Abstract: We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs give further evidence against the Linear Hirsch Conjecture. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3327-8

Authors:Nick Gill; Neil I. Gillespie; Jason Semeraro Abstract: Abstract To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M 13 which is constructed from P3. We show that Sp2m (2) and 22m . Sp2m (2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F2-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes. We also give a new characterization of M 13 and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3433-7

Authors:David Bevan Abstract: Abstract We prove that the set of growth rates of permutation classes includes an inFInite sequence of intervals whose inFImum is θ B ≈ 2:35526, and that it also contains every value at least θ B ≈ 2:35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ A ≈ 2:48187. Thus, we also refute his conjecture that the set of growth rates below λ A is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by Rényi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3349-2

Authors:John Bamberg; Melissa Lee; Koji Momihara; Qing Xiang Abstract: Abstract In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2) admitting \(C_{\left( {q^3 + 1} \right)/4} :C_3 \) . PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3525-4