found e.g. in [@diestel].

Given a graph $G$, we denote its vertex and edge sets by $V(G)$ and $E(G)$ respectively. For a vertex $v \in V(G)$, we write $N(v)$ for the neighbours of $v$ in $G$ and write $d(v):=|N(v)|$ for the degree of $v$. Given $S \subseteq V(G)$, we also write $d_S(v):= |N(v) \cap S|$ for the degree of $v$ in $S$. We denote with $\delta(G)$ the smallest degree among vertices in $G$.

We write $H \subseteq G$ to mean that $H$ is a subgraph of $G$, i.e. $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$. We define $E_G(S):=\{ab \in E(G) \mid a,b \in S \}$ and we write $G[S]$ for the graph induced by $G$ on $S$, i.e. the graph with vertex set $S$ and edge set $E_G(S)$. For $S,T\subseteq V(G)$, we define $E_G(S,T):=\{xy\in E(G)\mid x\in S, y\in T\}$ and $e_G(S,T):=|E_G(S,T)|$. We will sometimes omit the subscript if it is clear. For $S,T\subseteq V(G)$ disjoint, we write $G[S,T] := (S \cup T, E_G(S,T))$ for the bipartite graph induced by $G$ between $S$ and $T$. We often denote the complement of $S \subseteq V(G)$ by $\overline{S}$ i.e. $\overline{S}:= V(G) \setminus S$.

We write $a \ll b$ to mean that $a \leq 