The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering all the linearly dependent r-sets of $V = \mathbb{F}_{p^a}$ using $\Phi'$. This also covers "bounded" number of r-sets which have linearly independent entries(dimension r over $\mathbb{F}_{p}$)but does not cover any set from the template $G_r^{*}$.
-- Cover the r-sets of the template $G_r^*$ using $M(G^*)$. The template is algebraically defined and it basically is a refined version of all q-tuples of the form $My$ where the $q\times r$ matrix M has all the sub matrices of size r to be full rank and $y \in \mathbb{F}_{p^a}^r$.
-- Now cover the r-sets that are full rank, not in the template $G_r^*$ and not covered by $\Phi'$ using the nibble.
-- Now to cover the r-sets that are not covered by the nibble, edges are picked from $G_q$ (such that all r-sets except one belongs to $G_r^*$) in a greedy way randomly. This uses the processes defined in section 4 and generalised in section 5. This could cover few r-sets from the template $G_r^*$ again.
-- Express the sets of $G_r^*$ that are covered twice as the difference $\partial\Psi^+ - \partial\Psi^-$. This uses Lemma 5.28 which is a generalisation of Graver and Zurkat's result.
-- Apply local modifications on the covering of the template $M(G^*)$ to include $\partial\Psi^+$. This does not change $G_r^*$. This uses the cascade machinery built in the last subsection of section 5.
-- Replace $\partial\Psi^+$ with $\partial\Psi^-$ in the template cover and this completes the construction.
I was trying to understand the reduction as mentioned in the first step (Lemma 6.3). I do understand the way the complex is decomposed according to the intersection pattern with the set V' but I find myself stuck in the technicalities of the part where he proves that the greedy algorithm can always be completed(Page 46 last 2 paragraphs).
The exact places at which I am not very clear are:
1) The motivation for keeping track of the full sets in the reduction argument to $p^a$. (Lemma 6.3). I am not very sure as to where exactly this is used but I think this is used in getting the degree regularity requirements to apply the nibble on $A^x$ when $x^0 < r$.
However I do understand why full sets were needed to be kept track of in the initial elimination of the dependent sets after the reduction to p^a in the proof of Theorem 6.1(Page 48).
2) Also I am not very clear on the analysis of $G^2$ in the proof of Lemma 6.3(page 46). Where he analyses $G^2$ by repeatedly restricting to a random subset (p-binomial) and a bounded subset (ie using Lemma 3.19 repeatedly).
Could you please provide an intuitive explanation as to why the greedy algorithm here can be completed.
