I have been reading the paper 'Improved Bounds for the Sunflower Lemma' (Ann. of Math., Vol. 194(3), pp. 795-815), and have not managed to understand the following:
I would like a formalization for what a $p$-biased distribution (pg. 796) is, since $R$ is treated both as a set and a random-variable in Definition 1.5 (pg. 796) and I am not certain how this is done.
What I have tried: Consider the measure space $(X, \mathcal{P}(X), \mathbb{P})$ and the measurable space $(X, \mathcal{P}(X))$. We define the random-variable $\mathcal{R}: X \to X$ such that $\mathcal{R}: t \mapsto t$, and the probability-measure $\mathbb{P}: \mathcal{P}(X) \to [0, \infty]$, such that $\forall \, t \in X: \mathbb{P}(\{t\}) = p$, and $\forall \, S \subseteq X:$ $\mathbb{P}(S) := \prod_{s \in S} \mathbb{P}(\{s\}).$ Then, the distribution function is as follows- $\forall \, A \in \mathcal{P}(X):$ $$\mathbb{P}^{X}(A) = \mathbb{P}(\{w \in X: \mathcal{R}(w) \in A\}) = \mathbb{P}(A).$$ Is this indeed the formal definition of a $p$-biased distribution? If so, what does Definition 1.5 mean (, since $R$ is considered both as a r.v. and a set in the definition)? If not, what is a $p$-biased distribution formally?Assuming a heuristic understanding of $U(X,p)$ as being as stated by the authors on pg. 796 before Definition 1.5, we consider Lemma 1.6. On pg. 797, the paper claims that by 'the union bound', we have that $$\textrm{Pr} \: [\forall \, i \in [r], \, \exists \, S \in \mathcal{F}: S \subseteq Y_{i}] > 0$$, where $Y_{i}$ is the set of all points in $X$ coloured with colour $i$. I am not able to follow this, since I am not able to say anything non-trivial about $\textrm{Pr} (\cup_{i \in [r]} \varepsilon_{i})$.
When the authors define the weight function $\sigma$ on pg. 799, they restrict its domain to $\mathcal{F}$, but on pg. 800, they refer to $\sigma(\mathcal{F}_T)$, which doesn't make sense to me, since it doesn't appear that $\forall \, T \subseteq X, \, T \neq \emptyset: \mathcal{F}_T \subset \mathcal{F}.$ Therefore, I was wondering if we require that the domain of $\sigma$ be $\mathcal{P}(X)$ instead.
On pg. 800, after Definition 2.1, the authors state that if $(\mathcal{F},\sigma)$ is s-spread, where $s= (s_{0}; s_{1}, \dots, s_{w})$ is a weight-profile, in particular, $\mathcal{F}$ is a $w$-set-system. What I am able to show is that if indeed $\mathcal{F}$ contains a set $S$ with $|S| > w$, then, we have $$\forall R \in \mathcal{F}, S \subset R: \sigma (R \, \backslash \, S) = 0.$$ What I thought I could try showing was that $\sigma$ would be $0$ everywhere, which is not allowed, by definition of a weight function on pg. 799.
Any help on any or all of these questions is appreciated.