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Square-free values of polynomials had been studied by various authors, including Estermann, Heath-Brown and Hooley. For 1 ≤ x , y ≤ H , Tolev proved that the number of the square-free values attained by the polynomial x 2 + y 2 + 1 has the asymptotic formula c 1 H 2 + O ( H 4 / 3 + e ) , where is c 1 is an absolute constant and e is an arbitrary small positive number. The key ingredient of his proof which leads to the elaborate error term is the estimate for the Kloosterman sum. In this paper, by using Tolev's method and some estimate for the Salie sum, we show that for any fixed integer k, there is an absolute constant c 2 such that the number of square-free values of the polynomial x 2 + y 2 + z 2 + k with 1 ≤ x , y , z ≤ H is c 2 H 3 + O ( H 7 / 3 + e ) .

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching problem on a special hypergraph $H$. We then combine several existing techniques on matchings in uniform hypergraphs: find an absorbing matching $M$ in $H$; use a randomization process of Alon et al. to find an almost regular subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of {\L}uczak and Mieczkowska and might be of independent interest., Comment: added two references [2,7], accepted for publication in SCIENCE CHINA Mathematics

We consider the Massera-Schaffer problem for the equation $$ - y\prime (x) + q(x)y(x) = f(x),\,\,\,\,\,x \in \mathbb{R},$$ where $$f \in L_p^{{\rm{loc}}}(\mathbb{R})$$ , p ∈ [1, ∞) and $$0 \leqslant q \in L_p^{{\rm{loc}}}(\mathbb{R})$$ . By a solution of the problem we mean any function y, absolutely continuous and satisfying the above equation almost everywhere in $$\mathbb{R}$$ . Let positive and continuous functions μ(x) and θ(x) for $$x \in \mathbb{R}$$ be given. Let us introduce the spaces $$\matrix{{{L_p}(\mathbb{R},\mu ) = \left\{ {f \in L_p^{{\rm{loc}}}(\mathbb{R}):\left\| f \right\|_{{L_p}(\mathbb{R},\mu )}^p = \int_{ - \infty }^\infty {{{\left| {\mu (x)f(x)} \right|}^p}{\rm{d}}x < \infty } } \right\},} \hfill \cr {{L_p}(\mathbb{R},\theta ) = \left\{ {f \in L_p^{{\rm{loc}}}(\mathbb{R}):\left\| f \right\|_{{L_p}(\mathbb{R},\theta )}^p = \int_{ - \infty }^\infty {{{\left| {\theta (x)f(x)} \right|}^p}{\rm{d}}x < \infty } } \right\}.} \hfill \cr } $$ We obtain requirements to the functions μ θ and q under which (1) for every function $$f \in {L_p}(\mathbb{R},\theta )$$ there exists a unique solution $$y \in {L_p}(\mathbb{R},\mu )$$ of the above equation; (2) there is an absolute constant c(p) ∈ (0, ∞) such that regardless of the choice of a function $$f \in {L_p}(\mathbb{R},\theta )$$ the solution of the above equation satisfies the inequality $${\left\| y \right\|_{{L_p}(\mathbb{R},\mu )}} \leqslant c(p){\left\| f \right\|_{{L_p}(\mathbb{R},\theta ).}}$$ .

Let $$G$$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $$c$$ such that whenever $$S$$ is a non-trivial normal subset in $$G$$ then $$S^{k} = G$$ for any integer $$k$$ at least $$c \cdot (\log|G|/\log|S|)$$ . This result is generalized by showing that there exists an absolute constant $$c$$ such that whenever $$S_{1}$$ , $$\ldots , $$ $$S_{k}$$ are normal subsets in $$G$$ with $$\prod_{i=1}^{k} |S_{i}| \geq {|G|}^{c}$$ then $$S_{1} \cdots S_{k} = G$$ .

A well-known conjecture of Babai states that if G is a finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay ( G , X ) is bounded above by ( log ⁡ | G | ) c for some absolute constant c. The goal of this paper is to prove such a bound for the diameter of Cay ( G , X ) whenever G = S L ( n , p ) and X is a generating set of G which contains a transvection. A natural analogue of this result is also proved for G = S L ( n , K ) , where K can be any field.

In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 | A | p 2 , B ∩ − B = ∅ and | B | κ | A | ln ⁡ ( | A | ) , we have that max b ∈ B ⁡ | ( A + b ) ∖ A | ≥ | B | . In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed.

An example is presented of a subset $A$ of $\mathbb Z_N$ of density $\alpha$ such that the largest non-trivial Fourier coefficient of the characteristic function of $A$ is very small, but the probability that a random arithmetic progression (mod $N$) of length 4 lies in $A$ is significantly smaller than $\alpha^4$., Comment: This preprint was written well over a decade ago and has been known about by experts but not made publicly available. I apologize for this and am now remedying the situation. Revised version updates the notation, corrects minor errors (pointed out by referee), and adds one or two references

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

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we denote by $\mathcal{F}_{\Delta}$ the set all values $\gamma$ for which there exists an infinite subset $X\subset \mathbb{N}$ such that $X^{(2)}$ is $\gamma$-coloured. Properties of this set were first studied by Erickson in $1994$. Here, we are interested in estimating the minimum size of $\mathcal{F}_{\Delta}$ over all $m$-colourings $\Delta$ of $\mathbb{N}^{(2)}$. Indeed, we shall prove the following result. There exists an absolute constant $\alpha > 0$ such that for any positive integer $m \neq \left\{ {n \choose 2}+1, {n \choose 2}+2: n\geq 2\right\}$, $|\mathcal{F}_{\Delta}| \geq (1+\alpha)\sqrt{2m}$, for any $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, thus proving a conjecture of Narayanan. This result is tight up to the order of the constant $\alpha$., Comment: 22 pages

A $(k,\varepsilon)$-non-malleable extractor is a function ${\sf nmExt} : \{0,1\}^n \times \{0,1\}^d \to \{0,1\}$ that takes two inputs, a weak source $X \sim \{0,1\}^n$ of min-entropy $k$ and an independent uniform seed $s \in \{0,1\}^d$, and outputs a bit ${\sf nmExt}(X, s)$ that is $\varepsilon$-close to uniform, even given the seed $s$ and the value ${\sf nmExt}(X, s')$ for an adversarially chosen seed $s' \neq s$. Dodis and Wichs~(STOC 2009) showed the existence of $(k, \varepsilon)$-non-malleable extractors with seed length $d = \log(n-k-1) + 2\log(1/\varepsilon) + 6$ that support sources of entropy $k > \log(d) + 2 \log(1/\varepsilon) + 8$. We show that the foregoing bound is essentially tight, by proving that any $(k,\varepsilon)$-non-malleable extractor must satisfy the entropy bound $k > \log(d) + 2 \log(1/\varepsilon) - \log\log(1/\varepsilon) - C$ for an absolute constant $C$. In particular, this implies that non-malleable extractors require min-entropy at least $��(\log\log(n))$. This is in stark contrast to the existence of strong seeded extractors that support sources of entropy $k = O(\log(1/\varepsilon))$. Our techniques strongly rely on coding theory. In particular, we reveal an inherent connection between non-malleable extractors and error correcting codes, by proving a new lemma which shows that any $(k,\varepsilon)$-non-malleable extractor with seed length $d$ induces a code $C \subseteq \{0,1\}^{2^k}$ with relative distance $0.5 - 2\varepsilon$ and rate $\frac{d-1}{2^k}$., 14 pages, 1 figure

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path P n to be the monotone increasing path with n edges. The ordered size Ramsey number r ( P r , P s ) is the minimum number m for which there exists an ordered graph H with m edges such that every two-coloring of the edges of H contains a red copy of P r or a blue copy of P s . For 2 ≤ r ≤ s , we show 1 8 r 2 s ≤ r ( P r , P s ) ≤ C r 2 s ( log s ) 3 , where C > 0 is an absolute constant. This problem is motivated by the recent results of Bucic et al. (2019) and Letzter and Sudakov (2019) for oriented graphs.

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.

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