The probabilistic method

Colour the edges of $K_6$ red or blue by independent fair coin flips. What is the expected number of monochromatic triangles?

For a random colouring of $K_6$, the expected number $X$ of monochromatic $K_4$'s is $\binom 64 2^{1-6} = \frac{15}{32} \approx 0.47$. What can we conclude?

What is the largest $n$ for which the first-moment condition $\binom n4 \, 2^{1 - \binom 42} < 1$ holds? (It proves $R(4, 4) > n$.)

In the proof that every graph with $m$ edges has a cut of size $\ge m/2$, we write $X = \sum_e X_e$ with $X_e = 1$ if edge $e$ is cut, and conclude $\mathbb{E}[X] = m/2$. The $X_e$ are not mutually independent (the three edges of a triangle cannot all be cut). Why is the conclusion still valid?

In Erdős's proof of Theorem 3, with $p = n^{\theta - 1}$ and $\theta < 1/g$, why do we delete a vertex from each short cycle instead of asking that the random graph have no short cycles at all?

In a $10$-CNF formula (every clause has $10$ literals on distinct variables), each clause shares variables with at most $d$ other clauses. What is the largest $d$ for which the symmetric Lovász local lemma ($e\,p\,(d+1) \le 1$) guarantees that the formula is satisfiable?

Let $S$ be a sum of $n = 10000$ independent random signs $\pm 1$. Which is the Chernoff bound (Theorem 6) for $\mathbb{P}(S \ge 400)$?

With Colouring n the red/blue colourings of the edges of $K_n$ and Monochromatic c K meaning that all edges inside the vertex set K have the same colour under c, which Lean statement says $R(k, k) > n$?

Write expected_mono(n, k) returning $\binom nk 2^{1 - \binom k2}$ as an exact Fraction, and first_moment_bound(k) returning the largest $n \ge k$ with expected_mono(n, k) < 1 (so that $R(k, k) > n$).