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). Could you please provide an intuitive explanation as to why the greedy algorithm here can be completed.

Thank you

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.

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.

Now I was trying to understand the reduction as mentioned in the first step. 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). Could you please provide an intuitive explanation as to why the greedy algorithm here can be completed.

Thank you

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). Could you please provide an intuitive explanation as to why the greedy algorithm here can be completed.

Thank you