Authors:Karim A. Adiprasito; Arnau Padrol Pages: 129 - 136 Abstract: In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of a neighborly simplicial polytope. This in particular provides the final step for Mnëv‘s proof of the universality theorem for simplicial polytopes. PubDate: 2017-04-01 DOI: 10.1007/s00493-016-3253-9 Issue No:Vol. 37, No. 2 (2017)

Authors:Levent Alpoge Pages: 137 - 142 Abstract: Let L(n) be the number of Latin squares of order n, and let L even(n) and L odd(n) be the number of even and odd such squares, so that L(n)=L even(n)+L odd(n). The Alon-Tarsi conjecture states that L even(n) ≠ L odd(n) when n is even (when n is odd the two are equal for very simple reasons). In this short note we prove that $$\left {{L^{even}}\left( n \right) - {L^{odd}}\left( n \right)} \right \leqslant L{\left( n \right)^{\frac{1}{2} + o\left( 1 \right)}}$$ , thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Two proofs are given: both proceed by applying a differential operator to an exponential integral over SU(n). The method is inspired by a recent result of Kumar-Landsberg. PubDate: 2017-04-01 DOI: 10.1007/s00493-015-3373-7 Issue No:Vol. 37, No. 2 (2017)

Authors:Diana Ojeda-Aristizabal Pages: 143 - 155 Abstract: We give a constructive proof of the finite version of Gowers’ FIN k Theorem for both the positive and the general case and analyse the corresponding upper bounds provided by the proofs. PubDate: 2017-04-01 DOI: 10.1007/s00493-015-3223-7 Issue No:Vol. 37, No. 2 (2017)

Authors:Tai Do Duc; Bernhard Schmidt Pages: 167 - 182 Abstract: Let p be a prime and let A be a subset of F p with A = -A and A \ {0} ≤ 2log3(p). Then there is an element of F p which has a unique representation as a difference of two elements of A. PubDate: 2017-04-01 DOI: 10.1007/s00493-015-3282-9 Issue No:Vol. 37, No. 2 (2017)

Authors:Massimo Lauria; Pavel Pudlák; Vojtěch Rödl; Neil Thapen Pages: 253 - 268 Abstract: We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every c-Ramsey graph must contain a large subgraph with some properties typical for random graphs. PubDate: 2017-04-01 DOI: 10.1007/s00493-015-3193-9 Issue No:Vol. 37, No. 2 (2017)

Authors:David Perkinson; Qiaoyu Yang; Kuai Yu Pages: 269 - 282 Abstract: A depth-first search version of Dhar’s burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph solves a problem posed by R. Stanley. PubDate: 2017-04-01 DOI: 10.1007/s00493-015-3191-y Issue No:Vol. 37, No. 2 (2017)

Authors:Christine Bachoc; Oriol Serra; Gilles Zémor Abstract: A Theorem of Hou, Leung and Xiang generalised Kneser’s addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou’s conjecture. This result is a consequence of a strengthening of Hou et al.’s theorem that is inspired by an addition theorem of Balandraud and is obtained by combinatorial methods transposed and adapted to the extension field setting. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3529-0

Authors:Mauro Di Nasso; Maria Riggio Abstract: By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being x n + y m = z k with k ∉ {n, m}. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3640-2

Authors:Simao Herdade; John Kim; Swastik Koppartyy Abstract: We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets A, B of the finite field \(\mathbb{F}_p \) , the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset A + B in terms of the sizes of the sets A and B. Our theorem considers a general linear map \(L:\mathbb{F}_p^n \to \mathbb{F}_p^m \) , and subsets \(A_1 , \ldots A_n \subseteq \mathbb{F}_p\) , and gives a lower bound on the size of L(A 1 × A 2 × … × A n ) in terms of the sizes of the sets A 1, …, A n . Our proof uses Alon’s Combinatorial Nullstellensatz and a variation of the polynomial method. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3486-7

Authors:Matthias Hamann Abstract: We prove that the cut space of any transitive graph G is a finitely generated Aut(G)-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose cycle space is generated by cycles of bounded length is accessible. In addition, it implies Dunwoody’s conjecture that locally finite hyperbolic transitive graphs are accessible. As a further application, we obtain a combinatorial proof of Dunwoody’s accessibility theorem of finitely presented groups. PubDate: 2017-05-31 DOI: 10.1007/s00493-017-3361-1

Authors:Peter Frankl Abstract: Let H be a hypergraph of rank k, that is, H ≦ k for all H ∈ H. Let ν(H) denote the matching number, the maximum number of pairwise disjoint edges in H. For a vertex x let H(x̄) be the hypergraph consisting of the edges H ∈ H with x ∉ H. If ν(H(x̄)) = ν(H) for all vertices, H is called resilient. The main result is the complete determination of the maximum number of 2-element sets in a resilient hypergraph with matching number s. For k=3 it is \(\left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\) while for k ≧ 4 the formula is \(k \cdot \left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\) . The results are used to obtain a stability theorem for k-uniform hypergraphs with given matching number. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3579-3

Authors:Agnieszka Figaj; Tomasz Łuczak Abstract: We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-2433-y

Authors:Jacob Fox; László Miklós Lovász Abstract: We determine the order of the tower height for the partition size in a version of Szemerédi’s regularity lemma. This addresses a question of Gowers. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3274-4

Authors:Kolja Knauer; Luis Pedro Montejano; Jorge Luis Ramírez Alfonsín Abstract: Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attention to regular matroids as well as some graphic and cographic matroids leading in particular to the topics of (connected) bond and cycle covers in Graph Theory. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3556-x

Authors:Flavia Bonomo; Maria Chudnovsky; Peter Maceli; Oliver Schaudt; Maya Stein; Mingxian Zhong Abstract: In this paper we present a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and gives an explicit coloring if one exists. PubDate: 2017-05-31 DOI: 10.1007/s00493-017-3553-8

Authors:František Matúš Abstract: This work studies the classes of matroids that are closed under minors, addition of coloops and principal extensions. To any matroid M in such a class a matroid M° is constructed such that it contains M as a minor, has all proper minors in the class and violates Zhang-Yeung inequality. When the class enjoys the inequality the matroid M° becomes an excluded minor. An analogous assertion was known before for the linear matroids over any infinite field in connection with Ingleton inequality. The result is applied to the classes of multilinear, algebraic and almost entropic matroids. In particular, the class of almost entropic matroids has infinitely many excluded minors. PubDate: 2017-05-31 DOI: 10.1007/s00493-016-3534-y

Authors:Eli Berger 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