Authors:Iskander Aliev; Robert Bassett; Jesús A. De Loera; Quentin Louveaux Pages: 313 - 332 Abstract: Abstract The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedron {x∈R n : Ax≤b} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n,k), defining a polyhedron that contains exactly the same k integer points. In this case c(n,0)=2 n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant c(n,k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3266-9 Issue No:Vol. 37, No. 3 (2017)

Authors:Itai Benjamini; Hilary Finucane; Romain Tessera Pages: 333 - 374 Abstract: Abstract Let (X n ) be an unbounded sequence of finite, connected, vertex transitive graphs such that X n =O(diam(X n ) q ) for some q>0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (X n ) converges in the Gromov Hausdorff distance to some finite dimensional torus equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov’s theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If X n is only roughly transitive and X n =O diam(X n δ ) for δ >1 sufficiently small, we prove, this time by elementary means, that (X n ) converges to a circle. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-2975-4 Issue No:Vol. 37, No. 3 (2017)

Authors:Mauro Biliotti; Alessandro Montinaro Pages: 375 - 395 Abstract: Abstract Let П0 be a subplane of order q of PG(2,q 3) and let G be the copy of PGL(3,q) preserving П0. The Figueroa plane Fig(q 3) is constructed by replacing some parts of the lines of PG(2,q 3) external to П0 by suitable q-subgeometries of PG(2,q 3). Moreover, Fig(q 3) inherits G from PG(2,q 3). We show that this is the unique replacement for the external lines to П0 yielding a projective plane of order q 3 admitting G as a collineation group. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3337-y Issue No:Vol. 37, No. 3 (2017)

Authors:Pete L. Clark; Aden Forrow; John R. Schmitt Pages: 397 - 417 Abstract: Abstract We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning’s Second Theorem implies Chevalley’s Theorem, our result implies Schauz-Brink’s Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems. Let q = p ℓ be a power of a prime number p, and let F q be “the” finite field of order q. For a 1,...,a n , N∈Z+, we denote by m(a 1,...,a n ;N)∈Z+ a certain combinatorial quantity defined and computed in Section 2.1. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3267-8 Issue No:Vol. 37, No. 3 (2017)

Authors:Gil Cohen; Amir Shpilka; Avishay Tal Pages: 419 - 464 Abstract: Abstract We study the following problem raised by von zur Gathen and Roche [6]: What is the minimal degree of a nonconstant polynomial f: {0,..., n} → {0,..., m}' Clearly, when m = n the function f(x) = x has degree 1. We prove that when m = n — 1 (i.e. the point {n} is not in the range), it must be the case that deg(f) = n — o(n). This shows an interesting threshold phenomenon. In fact, the same bound on the degree holds even when the image of the polynomial is any (strict) subset of {0,..., n}. Going back to the case m = n, as we noted the function f(x) = x is possible, however, we show that if one excludes all degree 1 polynomials then it must be the case that deg(f) = n — o(n). Moreover, the same conclusion holds even if m=O(n 1.475-ϵ). In other words, there are no polynomials of intermediate degrees that map {0,...,n} to {0,...,m}. Furthermore, we give a meaningful answer when m is a large polynomial, or even exponential, in n. Roughly, we show that if \(m < (_{\,\,\,d}^{n/c} )\) , for some constant c, and d≤2n/15, then either deg(f) ≤ d—1 (e.g., \(f(x) = (_{\,\,\,d - 1}^{x - n/2} )\) is possible) or deg(f) ≥ n/3 - O(dlogn). So, again, no polynomial of intermediate degree exists for such m. We achieve this result by studying a discrete version of the problem of giving a lower bound on the minimal L ∞ norm that a monic polynomial of degree d obtains on the interval [—1,1]. We complement these results by showing that for every integer k = O( \(\sqrt n \) ) there exists a polynomial f: {0,...,n}→{0,...,O(2 k )} of degree n/3-O(k)≤deg(f)≤n-k. Our proofs use a variety of techniques that we believe will find other applications as well. One technique shows how to handle a certain set of diophantine equations by working modulo a well chosen set of primes (i.e., a Boolean cube of primes). Another technique shows how to use lattice theory and Minkowski’s theorem to prove the existence of a polynomial with a somewhat not too high and not too low degree, for example of degree n−Ω(logn) for m=n−1. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-2987-0 Issue No:Vol. 37, No. 3 (2017)

Authors:Natalia Iyudu; Stanislav Shkarin Pages: 465 - 479 Abstract: Abstract It follows from the Golod-Shafarevich theorem that if k ∈ N and R is an associative algebra given by n generators and $$d< \frac{{{n^2}}}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)$$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every k ∈ N, there is a sequence of algebras Rn given by n generators and dn quadratic relations such that R n is k-step nilpotent and $$\mathop {\lim }\limits_{n \to \infty } \frac{{{d_n}}}{{{n^2}}} = \frac{1}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)$$ . PubDate: 2017-06-01 DOI: 10.1007/s00493-016-3009-6 Issue No:Vol. 37, No. 3 (2017)

Authors:Alexandr Kostochka; Benny Sudakov; Jacques Verstraëte Pages: 481 - 494 Abstract: Abstract More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k≥k 0(ε) contains cycles of at least k 2−ε different lengths as k→∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k≥k 0(ε) contains cycles of 1/64(1 − ε)k 2 logk/4 consecutive lengths, and a cycle of length at least 1/4(1 − ε)k 2logk. As there exist triangle-free graphs of chromatic number k with at most roughly 4k 2 logk vertices for large k, these results are tight up to a constant factor. We also give new lower bounds on the circumference and the number of different cycle lengths for k-chromatic graphs in other monotone classes, in particular, for K r -free graphs and graphs without odd cycles C 2s+1. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3262-0 Issue No:Vol. 37, No. 3 (2017)

Authors:Lothar Narins; Alexey Pokrovskiy; Tibor Szabó Pages: 495 - 519 Abstract: Abstract We study graphs on n vertices which have 2n−2 edges and no proper induced subgraphs of minimum degree 3. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths 3,4,5,...,C(n) for some function C(n) tending to in finity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with n vertices and 2n−2 edges, containing no proper subgraph of minimum degree 3. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3310-9 Issue No:Vol. 37, No. 3 (2017)

Authors:Eszter Rozgonyi; Csaba Sándor Pages: 521 - 537 Abstract: Abstract For a given integer n and a set S ⊆ N denote by R h,S (1) the number of solutions of the equation \(n = {s_{{i_1}}} + ... + {s_{{i_h}}},{s_{{i_j}}} \in S,j = 1,...,h\) . In this paper we determine all pairs (A;B), A,B ⊆ N for which R h,A (1)(n) = R h,B (1)(n) from a certain point on, where h is a power of a prime. We also discuss the composite case. PubDate: 2017-06-01 DOI: 10.1007/s00493-015-3311-8 Issue No:Vol. 37, No. 3 (2017)

Authors:Terence Tao; Van Vu Pages: 539 - 553 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: 2017-06-01 DOI: 10.1007/s00493-016-3363-4 Issue No:Vol. 37, No. 3 (2017)

Authors:Christine Bachoc; Oriol Serra; Gilles Zémor Abstract: 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: 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: 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: 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: 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: 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: 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