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There is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with |λ| ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pal Turan.

We prove that there is an absolute constant A > 0 such that $$\begin{array}{l} \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}|P'(x)| \ge A\sqrt {n \cdot } \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}\,|P(x)| \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{array}$$ for an arbitrary algebraic polynomial P of degree n whose zeros lie in the half-disk $$\left\{ {z:|z| \le 1,{\mathop{\rm Im}\nolimits} z \ge 0} \right\}$$.

Let χ be a Dirichlet character modulo a prime p . We give explicit upper bounds on q 1 q 2 … q n , the n smallest prime nonresidues of χ . More precisely, given n 0 and p 0 there exists an absolute constant C = C ( n 0 , p 0 ) > 0 such that q n ≤ C p 1 4 ( log ⁡ p ) n + 1 2 whenever n ≤ n 0 and p ≥ p 0 .

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors $$p_{1},\ldots, p_{t}$$ of n and positive integers $$\alpha_{1}, \ldots , \alpha_{t}$$ such that $$n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}$$. Erdős and Hegyvari [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible by p. Recently, Fang and Chen [3] proved that for any given positive integer m, there are infinitely many weakly prime-additive numbers which are divisible bym with t = 3 if and only if $$8 \nmid m$$. In this paper, we prove that for any given positive integer m, the number of weakly prime-additive numbers which are divisible by m and less than x is larger than $${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$$ for all sufficiently large x, where c is a positive absolute constant. The constant c depends on the result on the least prime number in an arithmetic progression.

Let F be a self-dual Hecke-Maass cusp form for S L ( 3 , Z ) and let a F ( 1 , n ) denote the n-th coefficient of the Godement-Jacquet L-function L ( s , F ) . Then we show that there exists an absolute constant c 0 > 0 such that ∑ n ⩽ X a F ( 1 , n ) μ ( n ) e ( n α ) ≪ X exp ⁡ ( − c 0 log ⁡ X ) . Here the implied constant depends only on the form F and the bound is uniform in α ∈ R . Moreover, we notice that the aforementioned result generalises to self-dual automorphic cuspidal representations of G L 3 ( A Q ) , with unitary central character.

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way., 26 pages, no figures. Updated presentation

Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $\epsilon$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $\Theta(\log(1/\epsilon))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary., Comment: 18 pages

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0, Comment: Version 2: typos corrected and a new section on natural density added

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $\mathcal{G}$. We show that for every integer $d \geq 3$, almost every $d$-regular graph $G$ satisfies $\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$ for every ordering $\mathcal{G}$ of $G$. In particular, there are 3-regular graphs $G$ on $n$ vertices for which the numbers $\overline{R}(\mathcal{G})$ are superlinear in $n$, regardless of the ordering $\mathcal{G}$ of $G$. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph $G$ on $n$ vertices with maximum degree 2 admits an ordering $\mathcal{G}$ of $G$ such that $\overline{R}(\mathcal{G})$ is linear in $n$. We also show that almost every ordered matching $\mathcal{M}$ with $n$ vertices and with interval chromatic number two satisfies $\overline{R}(\mathcal{M}) \geq cn^2/\log^2{n}$ for some absolute constant $c$., 19 pages, 8 figures, minor corrections

Let θ 3 , 4 denote the graph consisting of three internally disjoint paths of four edges with the same pair of endpoints. In this paper, we give a lower bound of order n 5 / 4 on the greatest number of edges of any n-vertex θ 3 , 4 -free graph, matching an earlier upper bound by Faudree and Simonovits up to an absolute constant factor. The construction is algebraic in nature, arising from equations over finite fields, and is perhaps some evidence that the Turan Number for the octagon is also of order n 5 / 4 .

For any graph G = (V;E) and positive integer p, the exact distance-p graph G[\p] is the graph with vertex set V , which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[\p] is bounded by the weak (2p

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov., Appeared in SIAM J. Discrete Math. This version corrects some errors from the previous version, and clarifies the analysis

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer n > 1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by n^(c log log n), then G is virtually (log n)-step nilpotent., 15 pages, 1 figure. Accepted manuscript (to appear in Math. Ann.)

Journal of Combinatorial Theory. Series A, 187, ISSN:0097-3165, ISSN:1096-0899

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac {\log n}{\log \log n}$. This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd\H{o}s, Faudree, Rousseau and Schelp., Comment: 19 pages

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log \log |G:H|}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups., Comment: 6 pages

For an integer $m \geq 3$, does there exist an absolute constant $K(m)$ such that every polynomial with $m$ non-zero coefficients has an irreducible factor with at most $K(m)$ coefficients? A previous result in the literature establishes $K(3) \geq 9$, which is here improved to $K(3) \geq 12$. Improvements on known bounds are also given for $m=4,5,6$, and for $K(m)$, when $m \geq 7$.

O’Hara introduced several functionals as knot energies. One of them is the Mobius energy. We know its Mobius invariance from Doyle-Schramm’s cosine formula. It is also known that the Mobius energy was decomposed into three components keeping the Mobius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Mobius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Mobius invariant property. Using decomposition, the first and second variation formulae are derived.

The paper deals with a problem of Additive Combinatorics. Let G be a finite abelian group of order N . We prove that the number of subset triples A , B , C ⊂ G such that for any x ∈ A , y ∈ B and z ∈ C one has x + y ≠ z equals 3 ⋅ 4 N + N 3 N + 1 + O ( ( 3 − c ∗ ) N ) for some absolute constant c ∗ > 0 . This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and disproves the conjecture of Cohen et al. (0000). concerning the maximal possible number of independent sets in such hypergraphs on n vertices.

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\] where $\delta >0$ is an absolute constant only depending on $d$. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu., Comment: 14 pages

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$

Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>\lambda\}|\leq C\left(\frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}+\sup_{\nu} |\{|R_j\nu|>\lambda\}|\right) \end{align*} for any $\lambda>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $\nu=\sum_{k=1}^Na_k\delta_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16\|f\|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calder\'on-Zygmund operators., Comment: 17 pages

In the trace reconstruction problem, an unknown bit string ${\bf x}\in\{0,1 \}^n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in\{0,1 \}^n$ such that at least $c\, n^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c \log^2 n$ to $c \log^{9/4}n/\sqrt{\log \log n} $., Minor changes. 23 pages, 3 figures

BackgroundOptimal radii of curvature of the articulating surfaces of the prosthetic components are factors associated with the longevity of cemented glenoid components in anatomical total shoulder arthroplasty. It was the purpose of this study, to evaluate the radiographic and clinical performance of an anatomical glenoid component of a total shoulder arthroplasty (TSA) with respect to radial mismatch of the glenoid and humeral component.MethodsIn a retrospective study 75 TSA were analyzed for their clinical and radiographic performance with computed tomography by independent examiners using an established methodology. The study group was divided in two groups, one with mismatch ResultsThe mean glenohumeral radial mismatch was 3.4 mm (range 0.5–6.9). At median follow-up of 41 months (range 19–113) radiographic loosening (defined as modified Molé scores ≥6) was present in 7 cases (9.3%). Lucencies around the glenoid pegs (defined as modified Molé score ≥ 1) were present in 34 cases (45%). Radiolucencies were significantly associated with a radial mismatch p = 0.000). The pre- to postoperative improvements in Subjective Shoulder Value and absolute Constant Score were significantly better in the group with a mismatch ≥4.5 mm (p = 0.018,p = 0.014).ConclusionA lower conformity of the radii of humerus and glenoid seems to improve the loosening performance in TSA. Perhaps cut-off values regarding the recommended mismatch need to be revalued in the future.

Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tilde{\Lambda}(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tilde{\Lambda}(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tilde{\Lambda}(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tilde{\Lambda}(K)\leqslant(CL_K)^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.

A problem in zero-sum theory is to determine all pairs $(k,n)$ for which every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While all other cases have been solved more than a decade ago, the case when $k$ equals $4$ and $n$ is coprime to $6$ remains open. Precisely, The Index Conjecture in this subject states that if $n$ is coprime to $6$ then every minimal zero-sum sequence of length $4$ modulo $n$ has index $1$. In this paper, we prove an equivalent version of this conjecture for all $n>N$ for some absolute constant $N$.

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials., Inaccuracies corrected following the referee's suggestions

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.

We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset \{0,1\}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. This improves upon the $Cn^{3.75}$ bound of Gupta, Lee and Li and is sharp up to the value of the constant., 11 pages

We give quantitative generalized Voronovskaja’s formulae for Bernstein polynomials which are simple to compute. To achieve this, we obtain explicit and accurate upper estimates for the even central moments of Bernstein polynomials. As a consequence, we improve the value of the absolute constant in Videnskiĭ type inequalities.

We show that there exists an absolute constant A such that for each k ≥ 2 and every coloring of the edges of the complete k -uniform hypergraph on A r vertices with r colors, r ≥ r k , one of the color classes contains a loose path of length three.

Let p be a prime number and let q = p r . If C and D are large subsets of F q ⁎ we study the trace of products cd with c ∈ C and d ∈ D and show that it is well distributed in F p . We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in F p . We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k-th powers for k ≥ 1 , respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below q . Character sums and Gaussian sums over F p and F q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields.