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The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof. [ABSTRACT FROM AUTHOR]

Abstract: [E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the -Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon''s theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result. [Copyright &y& Elsevier]

Abstract: Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented. [Copyright &y& Elsevier]

Abstract: In a previous paper, we studied the cosets of weight 4 of binary extended 3-error-correcting BCH codes of length (where m is odd). We expressed the number of codewords of weight 4 in such cosets in terms of exponential sums of three types, including the Kloosterman sums , . In this paper, we derive some congruences which link Kloosterman sums and cubic sums. This allows us to study the divisibility of Kloosterman sums modulo 24. More precisely, if we know the traces of a and of , we are able to evaluate modulo 24 and to compute the number of those a giving the same value of modulo 24. [Copyright &y& Elsevier]

Abstract: In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the ▪-diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319–342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz''q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332–350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319–342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot. [Copyright &y& Elsevier]

Abstract: Recent work of Gowers [T. Gowers, A new proof of Szemerédi''s theorem, Geom. Funct. Anal. 11 (2001) 465–588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299–345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275–291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. [Copyright &y& Elsevier]

Abstract: Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand–Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or “hives”) on a “convex part” of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applications are discussed. In particular, we characterize, in terms of valid inequalities, the polyhedral cone formed by the boundary values of discrete strip-concave functions on a grid having trapezoidal configuration. As a consequence of this result, necessary and sufficient conditions on a pair of vectors to be the shape and content of a semi-standard skew Young tableau are obtained. [Copyright &y& Elsevier]

Abstract: This paper is motivated by Ito''s paper ((Hokkaido) Math. J. 22 (1993) 373), in which he constructed -relative difference sets in . Chen and Li have shown that , the alternating group of degree 5, has a (30,2,29,14)-relative difference set (see Discrete Math. 256 (2002) 301). These are the only examples of nontrivial relative difference sets in nonsolvable groups known to the author. In this paper, we study relative difference sets in and show that there exist exactly two nontrivial relative difference sets in up to equivalence with the aid of a computer. [Copyright &y& Elsevier]

Abstract: In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space R , whose chromatic number is a positive integer in the Zermelo–Fraenkel-choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory. [Copyright &y& Elsevier]

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q , t -Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free. [ABSTRACT FROM AUTHOR]

This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement. [ABSTRACT FROM AUTHOR]

In this paper, we give an algebraic construction of a new infinite family of Cameron–Liebler line classes with parameter x = q 2 − 1 2 for q ≡ 5 or 9 ( mod 12 ) , which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG ( 2 , q ) , which is closely related to our Cameron–Liebler line classes. [ABSTRACT FROM AUTHOR]

Given a k × l ( 0 , 1 ) -matrix F , we denote by fs ( m , F ) the largest number for which there is an m × fs ( m , F ) ( 0 , 1 ) -matrix with no repeated columns and no induced submatrix equal to F . A conjecture of Anstee, Frankl, Füredi and Pach states that, regarding F as fixed, fs ( m , F ) = O ( m k ) . The main results of this paper are that for k = 2 , fs ( m , F ) = O ( m 2 + o m ( 1 ) ) and for k ≥ 3 that fs ( m , F ) = O ( m ( 5 / 3 ) k − 1 + o m ( 1 ) ) . [ABSTRACT FROM AUTHOR]

Abstract: Let be a union-closed family of subsets of an m-element set A. Let and for let denote the number of sets in that contain a. Frankl''s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element with . Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280–288] we verify the conjecture for the particular case when and . Moreover, for these “large” families we prove an even stronger version via averaging. Namely, the sum of the , for all , is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices. [Copyright &y& Elsevier]

Abstract: Let G be a block-transitive automorphism group of a 2- design . It has been shown that the pairs fall into three classes: those where G is unsolvable and is flag-transitive, those where G is a subgroup of , and those where G is solvable and is of small order. Not much is known about the latter two classes. In this paper, we investigate the existence of 2- designs admitting a block-transitive automorphism group . Using Weil''s theorem on character sums, the following theorem is proved: if a prime power q is large enough and then there is a 2- design which has a block-transitive, but nonflag-transitive automorphism group G. Moreover, using computers, some concrete examples are given when q is small. [Copyright &y& Elsevier]

Abstract: In this paper we study the extremal problem of finding how many 1 entries an n by n 0–1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0–1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0–1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0–1 matrix) linear if its extremal function is . Our main results are modest steps towards the elusive goal of characterizing linear patterns. We find novel ways to generate new linear patterns from known ones and use this to prove the linearity of some patterns. We also find the first minimal non-linear pattern of weight above 4. We also propose an infinite sequence of patterns that we conjecture to be minimal non-linear but have as their extremal function. We prove a weaker statement only, namely that there are infinitely many minimal not quasi-linear patterns among the submatrices of these matrices. For the definition of these terms see below. [Copyright &y& Elsevier]

Abstract: In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and Méndez, and express the species of connected graphs as the exponential composition of the species of prime graphs. [Copyright &y& Elsevier]

Abstract: Hanaki [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195–201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363–369] generalized many properties of characters of finite groups to characters of association schemes. In this paper we show that many of these properties also hold for table algebras. Our approach is not to generalize the proofs in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195–201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363–369] to table algebras, but to prove many stronger properties, and then obtain results in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195–201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363–369] as direct consequences. [Copyright &y& Elsevier]

Abstract: In this paper we obtain a new description of the translation dual of a semifield introduced in [G. Lunardon, Translation ovoids, J. Geom. 76 (2003) 200–215]. Using such a description we are able to prove that a semifield and its translation dual have nuclei of the same order. Combining the Knuth cubical array and the translation dual, we give an alternate description of the chain of twelve semifields in the table of [S. Ball, G.L. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (2007) 117–129]. [Copyright &y& Elsevier]

Abstract: It is proved that every Latin square has a partial transversal of length at least . The previous papers proving these results (including one by the second author) not only contained an error, but were sloppily written and quite difficult to understand. We have corrected the error and improved the clarity. [Copyright &y& Elsevier]

Abstract: The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional equation for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by , and an asymmetric wedge defined by the lines and , where is an integer. We prove that the growth constant for all these models is equal to , independent of the angle of the wedge. We derive function equations for both models, and obtain explicit expressions for the generating functions when . From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when . The functional equations are solved by a variation of the kernel method, which we call the “iterated kernel method.” This method appears to be similar to the obstinate kernel method used by Bousquet-Mélou (see, for example, references [M. Bousquet-Mélou, Counting walks in the quarter plane, in: Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Trends Math., Birkhäuser, 2002, pp. 49–67; M. Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels, Electron. J. Combin. 9 (2) (2003) R19; M. Bousquet-Mélou, M. Petkovšek, Walks confined in a quadrant are not always D-finite, Theoret. Comput. Sci. 307 (2) (2003) 257–276]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θ-functions, and have natural boundaries on the unit circle. [Copyright &y& Elsevier]

Abstract: The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms. [Copyright &y& Elsevier]

Abstract: Let G be a finite abelian group of order n and let be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by , to be the least natural number k such that for any sequence with , there exists a non-empty subsequence and such that . Similarly, for any such set A, is defined to be the least such that for all sequences with , there exist indices , and with . In the present paper, we establish a relation between the constants and under certain conditions. Our definitions are compatible with the previous generalizations for the particular group and the relation we establish had been conjectured in that particular case. [Copyright &y& Elsevier]

Abstract: Separating hash families are useful combinatorial structures which are studied in a general form in this paper. Necessary and sufficient conditions for the existence of certain types of generalized hash functions are considered. [Copyright &y& Elsevier]

Abstract: Let G be a cyclic group of order and a sequence over G. We say that S is a zero-sum sequence if and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence.The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let be the smallest integer such that every minimal zero-sum sequence S over G with length satisfies . In this paper, we first prove that for . Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences. [Copyright &y& Elsevier]

Abstract: By studying the reciprocity property of linear Diophantine systems in light of Malcev–Neumann series, we present in this paper a new approach to and a generalization of Stanley''s monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley''s reciprocity theorem for linear homogeneous Diophantine systems. [Copyright &y& Elsevier]

Abstract: Given any linear code C over a finite field we show how C can be described in a transparent and geometrical way by using the associated Bruen–Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. ] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060–1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in , MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1–3) (1990) 1–11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441–459]. [Copyright &y& Elsevier]

Abstract: This paper concerns the embedding problem for partial totally symmetric quasigroups. For all , it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and , and this is the best possible such inequality. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the largest maximal partial spreads of the hermitian variety consist of generators. Previously, it was only known that is an upper bound for the size of these partial spreads. We also show for that every maximal partial spread of contains at least planes. Previously, only the lower bound was known. [Copyright &y& Elsevier]

Abstract: This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon''s approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role. [Copyright &y& Elsevier]

Abstract: Answering a question of Körner and Simonyi, this paper gives a strongly consecutive repeat-free code of maximal size in . [Copyright &y& Elsevier]

Abstract: Let denote the number of unrestricted partitions of n. For , 2, let denote the number of partitions of n such that . Here denotes the number of odd parts of the partition and is the conjugate of . Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of . Recently, Swisher [The Andrews–Stanley partition function and , preprint, submitted for publication] employed the circle method to show thatand that for sufficiently large n In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi''s triple product identity and some naive upper bound estimates. [Copyright &y& Elsevier]

Abstract: If and are linear equations, then the disjunctive Rado number of the set is the least integer n, provided that it exists, such that for every 2-coloring of the set there exists a monochromatic solution to either or . If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and , the disjunctive Rado number for the equations and is if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers and , the disjunctive Rado number for the equations and is if there exist natural numbers , and t such that and and is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise. [Copyright &y& Elsevier]

Abstract: Let be the group of monomial matrices, i.e., the group generated by all permutation matrices and diagonal matrices in . The group acts on the set of all subspaces of . The number of orbits of this action, denoted by , is the number of non-equivalent linear codes in . It was conjectured by Lax that as . We confirm this conjecture in this paper. [Copyright &y& Elsevier]

Abstract: Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons,preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space. [Copyright &y& Elsevier]

Abstract: We say that a 0–1 matrix A avoids another 0–1 matrix (pattern) P if no matrix obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17–27; J. Combin. Theory Ser. A 55 (1990) 316–320; Discrete Math. 103 (1992) 233–251) and other papers we investigate n by n 0–1 matrices avoiding a pattern P and the maximal number of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered. [Copyright &y& Elsevier]

Abstract: In (Adv. Math. 174(2) (2003) 236), a bijection between collections of reduced factorizations of elements of the symmetric group was described. Initially, this bijection was used to show the Schur positivity of the Stanley symmetric functions. Further investigations have revealed that our bijection has strong connections to other more familiar combinatorial algorithms. In this paper we will show how the Robinson–Schensted–Knuth correspondence can be decomposed into a sequence of applications of this bijection. [Copyright &y& Elsevier]

Abstract: The method of graphical condensation for enumerating perfect matchings was found by Propp (Theoret. Comput. Sci. 303 (2003) 267), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004) 29). In this paper, we obtain some more general results on graphical condensation than Kuo''s. Our method is also different from Kuo''s. As applications of our results, we obtain a new proof of Stanley''s multivariate version of the Aztec diamond theorem and we enumerate perfect matchings of a type of molecular graph. Finally, a combinatorial identity on the number of plane partitions is also given. [Copyright &y& Elsevier]

Abstract: Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that can never be 7, and that it is always of the form (q a prime power) for solvable groups G. In this paper we give exact or asymptotic formulas for . In particular, we show that , while for alternating groups we find unless or 9. An application of this result is also given. [Copyright &y& Elsevier]

This paper answers a query of Vincze (Acta Univ. Szeged, Sect. Sci. Math. 12 A (1950) 136–142): find the convex octagon with unit-length sides and minimum diameter. It also shows that the solution is #x03B5;-unique. The proof uses geometric arguments and a global optimization algorithm to solve a nonconvex quadratic program. [Copyright &y& Elsevier]

In this paper, subquadrangles of order s of generalized quadrangles (GQs) of order (s,s2) are investigated. In the case where S is a dual flock GQ of order (s,s2), s even, the classification of subquadrangles of order s is completed. In the case where E is an egg good at an element and the corresponding translation generalized quadrangle T(E) of order (s,s2), s even, has base point (∞) it is proved that if T(E) has a subquadrangle of order s not containing (∞), then T(E) is the classical GQ Q(5,s). Also a new strong characterization of the classical GQ Q(5,s), s even or odd, is obtained. In Part II, the case s odd is considered. [Copyright &y& Elsevier]

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using F-partitions. In this paper, Ramanujan''s 1ψ1 summation and the q-Gauss summation are established combinatorially. [Copyright &y& Elsevier]

In this paper we introduce the notion of s-extremal codes for self-dual binary codes and we relate this notion to the existence of 1-designs or 2-designs in these codes. We extend the classification of codes with long shadows of Elkies (Math. Res. Lett. 2(5) (1995) 643) to codes with minimum distance 6, for which we give partial classification. [Copyright &y& Elsevier]

We give a classification of all the countable, 1-transitive, coloured linear orderings for countable colour sets. This is a generalization of Morel''s classification of the countable, 1-transitive linear orderings. For finite colour sets, there are ℵ1 examples and for countably infinite colour sets, there are 2ℵ0 (discussed in more detail in a subsequent paper (countable, 1-transitive, coloured linear orderings II, submitted)). We also include a classification of the countable homogeneous coloured linear orders. [Copyright &y& Elsevier]

Let C be a class of relational structures. We denote by fC(n) the number of structures in C over the labeled set {0,…,n−1}. For any C definable in monadic second-order logic with unary and binary relation symbols only, E. Specker and C. Blatter showed that for every m∈N, the function fC satisfies a linear recurrence relation modulo m, and hence it is ultimately periodic modulo m. The case of ternary relation symbols, and more generally of arity k symbols for k⩾3, was left open.In this paper we show that for every m there is a class of structures Cm, which is definable even in first-order logic with one quaternary (arity four) relation symbol, such that fCm is not ultimately periodic modulo m. This shows that the Specker–Blatter Theorem does not hold for quaternary relations, leaving only the ternary case open. [Copyright &y& Elsevier]