As usual, $R_{k}^{(r)}(m)$ denotes the smallest integer $n$ such that every $k$-colouring of $[n]^{(r)}$ yields a monochromatic $m$-subset $M \in[n]^{(m)}$. Prove that $R_{2}^{(2)}(m)>2^{m / 2}$ for $m \geqslant 3$.

Let $\mathcal{P}([n])$ have the colex order, and for $a, b \in \mathcal{P}([n])$ let $\delta(a, b)=\max a \triangle b$; thus $a<b$ means $\delta(a, b) \in b$. Show that if $a<b<c$ then $\delta(a, b) \neq \delta(b, c)$, and that $\delta(a, c)=\max \{\delta(a, b), \delta(b, c)\} .$

Given a red-blue colouring of $[n]^{(2)}$, the 4 -colouring

$$\chi: \mathcal{P}([n])^{(3)} \rightarrow\{\text { red, blue }\} \times\{0,1\}$$

is defined as follows:

$$\chi(\{a, b, c\})= \begin{cases}(\text { red, } 0) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is red and } \delta(a, b)<\delta(b, c) \\ (\text { red }, 1) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is red and } \delta(a, b)>\delta(b, c) \\ (\text { blue, } 0) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is blue and } \delta(a, b)<\delta(b, c) \\ \text { (blue, } 1) & \text { if }\{\delta(a, b), \delta(b, c)\} \text { is blue and } \delta(a, b)>\delta(b, c)\end{cases}$$

where $a<b<c$. Show that if $M=\left\{a_{0}, a_{1}, \ldots, a_{m}\right\} \in \mathcal{P}([n])^{(m+1)}$ is monochromatic then $\left\{\delta_{1}, \ldots, \delta_{m}\right\} \in[n]^{(m)}$ is monochromatic, where $\delta_{i}=\delta\left(a_{i-1}, a_{i}\right)$ and $a_{0}<a_{1}<\cdots<a_{m}$.

Deduce that $R_{4}^{(3)}(m+1)>2^{2^{m / 2}}$ for $m \geqslant 3$.