General

• $f=O(g)$ or $f\ll g$ means there exists some absolute constant $C>0$ such that $\lvert f(x)\rvert\leq Cg(x)$ for all sufficiently large $x$. If the constant depends on some parameter this is indicated by subscripts, such as $f=O_k(g)$.
• $f=o(g)$ means for all $\epsilon>0$ we have $\lvert f(x)\rvert\leq \epsilon g(x)$ for all sufficiently large $x$.
• A sequence $a_1\leq a_2\leq \cdots$ is lacunary if there exists some $\lambda > 1$ such that $a_{i+1}\geq \lambda a_i$ for all $i\geq 1$.

Combinatorics and Graph Theory

• If $n\geq 1$ and $G$ is a finite graph then $\mathrm{ex}(n;G)$ is the maximum number of edges that a graph on $n$ vertices without $G$ as a subgraph can contain.
• If $k\geq 1$ then $R(k)$ is the smallest $N$ such that any 2-colouring of the edges of $K_N$ contains a monochromatic $K_k$.
• More generally, for any finite graphs $G,H$ we write $R(G,H)$ for the smallest $N$ such that any $2$-colouring of the edges of $K_N$ contains either a red $G$ or a blue $H$. If $G=H$ we just write $R(G)$.
• Let $r\geq 2$. The hypergraph Ramsey number $R_r(k)$ is the smallest $N$ such that any 2-colouring of the edges of the $r$-uniform hypergraph on $N$ vertices contains a monochromatic copy of the complete $r$-uniform hypergraph on $k$ vertices.
• The following definition is common in infinite Ramsey theory: let $\alpha,\gamma$ be ordinals and for each $\xi<\gamma$ let $\beta_\xi$ be an ordinal. Let $r\geq 1$ be a natural number. We write \[\alpha \to (\beta_\xi)_\gamma^r\] to mean that in any $\gamma$-colouring of $[\alpha]^r$ there exists some $\xi<\gamma$ and $X\subseteq \alpha$ such that the order type of $X$ is $\beta_\xi$ and all elements of $X^r$ receive the same colour. For example, the fact that in any $2$-colouring of $K_6$ there must exist a monochromatic $K_3$ would be written as $6 \to (3,3)_2^2$. We may omit the subscript $\gamma$ where it is clear from context. We may also write some non-complete hypergraph $G_\xi$ in place of the $\beta_\xi$.

Number Theory

• The (natural) density of a set $A\subseteq \mathbb{N}$ is \[d(A)=\lim_{N\to \infty}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N},\] when this limit exists. The upper and lower density are this with $\lim$ replaced by $\limsup$ and $\liminf$ respectively. The Schnirelmann density $d_s(A)$ is this with $\lim$ replaced by $\inf$.
• A Sidon set is a set such that the only solutions to $a+b=c+d$ are when $\{a,b\}=\{c,d\}$.
• If $f,g$ are functions then \[f\ast g(n) = \sum_{a+b=n}f(a)g(b)\] and \[f\circ g(n) = \sum_{a-b=n}f(a)\overline{g(b)}.\] In particular if $A$ is a set then $1_A\ast 1_A(n)$ counts the number of representations of $n$ as a sum of elements from $A$ and $1_A\circ 1_A(n)$ the number of representations as a difference.
• A set $A\subseteq \mathbb{N}$ is an additive basis of order $r$ if all sufficiently large integers are the sum of at most $r$ integers from $A$.
• For $A\subseteq\mathbb{N}$ we write \[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}.\] If $P(A)$ contains all sufficiently large integers then $A$ is complete. If $P(A)$ contains an infinite arithmetic progression then $A$ is subcomplete. If $A\backslash B$ is complete for any finite set $B$ then $A$ is strongly complete.