Extremal and additive combinatorics

How many edges does the Turán graph $T(10, 3)$ have? (By Turán's theorem, this is the maximum number of edges of a $K_4$-free graph on $10$ vertices.)

What is the maximum number of edges in a triangle-free graph on $11$ vertices?

What upper bound does the Erdős–Szekeres inequality $R(s, t) \le \binom{s + t - 2}{s - 1}$ give for $R(4, 5)$?

In the Zykov symmetrisation proof of Turán's theorem, we replace a vertex $u$ by a copy $v'$ of a vertex $v$ with $u \not\sim v$ (so $v'$ is adjacent exactly to $N(v)$). Why does the new graph still contain no $K_{r+1}$?

Szemerédi's theorem assumes positive upper density. Which infinite set shows that being infinite is not enough to force even a three-term progression?

What exactly does the Green–Tao theorem say?

In the proof of Lemma 7 (Roth's key lemma) we bounded $\left|\sum_{r \neq 0} \hat f(r)^2 \hat f(-2r)\right| \le \alpha \max_{s \neq 0}|\hat f(s)|$. Which facts does this step use?

Recall RamseyProp α s t m: in every graph on α, every vertex set W with m ≤ W.card contains an s-clique or an independent t-set. Which statement expresses $R(3, 3) \le 6$, and which theorem of the lesson proves it?

Dvir's theorem says that a Kakeya set $K \subseteq \mathbb{F}_q^n$ has $|K| \ge \binom{q + n - 1}{n}$. What lower bound does it give for a Kakeya set in $\mathbb{F}_5^2$?

Write three_ap_free(A) (does the set A of integers contain no $a \lt b \lt c$ with $a + c = 2b$?) and roth_number(N), the largest size of a 3-AP-free subset of range(N), by trying all subsets from the largest size down. Check $r_3(N)$ for $N \le 12$.