Let $n\ge 2k\ge 4$ be integers and $\mathcal{ℱ}$ a family of $k$-subsets of $\left\{1,2,\dots ,n\right\}$. We call $\mathcal{ℱ}$ intersecting if $F\cap F^{\prime }\ne \varnothing$ for all $F,F^{\prime }\in \mathcal{ℱ}$, and we call $\mathcal{ℱ}$ nontrivial if ${\bigcap }_{F\in \mathcal{ℱ}}F=\varnothing$. Strengthening the famous Erdős–Ko–Rado theorem, Hilton and Milner proved that $\left|\mathcal{ℱ}\right|\le \left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0.0pt}{}{n-k-1}{k-1}\right)+1$ if $\mathcal{ℱ}$ is nontrivial and intersecting. We provide a proof by injection of this result.

Let ${\mathbb{F}}_q$ be a finite field of order $q$. We prove that if $d\ge 2$ is even and $E\subset {\mathbb{F}}_q^d$ with $\left|E\right|\ge 9q^{d∕2}$ then

$${\mathbb{F}}_q=\frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)}=\left\{\frac{a}{b}:a\in \mathrm{\Delta }\left(E\right),b\in \mathrm{\Delta }\left(E\right)\setminus \left\{0\right\}\right\},$$

where

$$\mathrm{\Delta }\left(E\right)=\left\{\parallel x-y\parallel :x,y\in E\right\},\parallel x\parallel =x_1^2+x_2^2+\cdots +x_d^2.$$

If the dimension $d$ is odd and $E\subset {\mathbb{F}}_q^d$ with $\left|E\right|\ge 6q^{d∕2}$, then

$$\left\{0\right\}\cup {\mathbb{F}}_q^{+}\subset \frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)},$$

where ${\mathbb{F}}_q^{+}$ denotes the set of nonzero quadratic residues in ${\mathbb{F}}_q$. Both results are, in general, best possible, including the conclusion about the nonzero quadratic residues in odd dimensions.

Many recent results in combinatorics concern the relationship between the size of a set and the number of distances determined by pairs of points in the set. One extension of this question considers configurations within the set with a specified pattern of distances. In this paper, we use graph-theoretic methods to prove that a sufficiently large set $E$ must contain at least $C_G\left|E\right|$ distinct copies of any given weighted tree $G$, where $C_G$ is a constant depending only on the graph $G$.

An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to prove the identity.

We give a Matiyasevich-type identity for hypergeometric Bernoulli polynomials and their generalizations. By using this identity, we also give an identity for poly-Bernoulli polynomials.

We prove that if $g\left(x,y\right)$ is a polynomial of degree $d$ that is not a polynomial of only $y$, then for any finite set $A\subset ℝ$

$$\left|X\right|{\gg }_d|A{|}^2,\text{ where }X:=\left\{\frac{g\left(a_1,b_1\right)-g\left(a_2,b_2\right)}{b_2-b_1}:a_1,a_2,b_1,b_2\in A\right\}.$$

We will see this bound is also tight for some polynomial $g\left(x,y\right)$.

For a finite abelian group the Lind–Lehmer constant is the minimum positive logarithmic Lind–Mahler measure for that group. Finding this is equivalent to determining the minimal nontrivial group determinant when the matrix entries are integers.

For a group of the form $G={\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$ with $\left|G\right|\ge 4$ we show that this minimum is always $\left|G\right|-1$, a case of sharpness in the trivial bound. For $G={\mathbb{Z}}_2\times {\mathbb{Z}}_{2^n}$ with $n\ge 3$ the minimum is $9$, and for $G={\mathbb{Z}}_3^{}\times {\mathbb{Z}}_3^n$ the minimum is $8$. Previously the minimum was only known for $2$- and $3$-groups of the form $G={\mathbb{Z}}_p^k$ or ${\mathbb{Z}}_{p^k}$. We also show that a congruence satisfied by the group determinant when $G={\mathbb{Z}}_p^r$ generalizes to other abelian $p$-groups.

We construct a sequence of lattices $\left\{L_{n_i}\subset {ℝ}^{n_i}\right\}$ for $n_i\to \infty$ with exponentially large kissing numbers, namely, ${log}_2\tau \left(L_{n_i}\right)>0.0338\cdot n_i-o\left(n_i\right)$. We also show that the maximum lattice kissing number ${\tau }_n^l$ in $n$ dimensions satisfies ${log}_2{\tau }_n^l>0.0219\cdot n-o\left(n\right)$ for any $n$.

Let $p$ be a large enough prime number. When $A$ is a subset of ${\mathbb{F}}_p\backslash \left\{0\right\}$ of cardinality $\left|A\right|>\left(p+1\right)∕3$, then an application of the Cauchy–Davenport theorem gives ${\mathbb{F}}_p\backslash \left\{0\right\}\subset A\left(A+A\right)$. In this note, we improve on this and we show that $\left|A\right|\ge 0.3051p$ implies $A\left(A+A\right)\supseteq {\mathbb{F}}_p\backslash \left\{0\right\}$. In the opposite direction we show that there exists a set $A$ such that $\left|A\right|>\left(\frac{1}{8}+o\left(1\right)\right)p$ and ${\mathbb{F}}_p\backslash \left\{0\right\}⊈A\left(A+A\right)$.

We give a short proof that for each multiplicative subgroup $H$ of finite index in ${\mathbb{Q}}^{+}$, the set of integers $a$ with $a,a+1\in H$ is an IP-set. This generalizes a theorem of Hildebrand concerning completely multiplicative functions taking values in the $k$-th roots of unity.