Distributing shares in a particular monotone access structure Consider 6 people, $A,B,C,D,E,F$ and a secret. Construct a scheme which enables the following subsets of people to retrieve the secret: 

three players from the set $\{A,B,C,D\}$

two players from the set $\{A,E,F\}$

two players
from the set $\{B,C,D\}$ together with at least one player from the set $\{A,E, F\}$.

I want to know how I can distribute $6$ shares amongst these people so that the secret can only be obtained when three shares come together i.e a threshold of three. I've tried so many options but there always seems to be one unauthorised subset that gains access to the secret that causes it not to work. For example if I label the shares $1,2,3,4,5,6$ and distribute them like the following:

$A - 1$

$B- 2$

$C - 3$

$D - 6$

$E - 2,5$

$F - 3,4$

Then the unauthorised set $\{D,F\}$ for example will have three shares and be able to recover the secret. Is there a way to distribute them so that only authorised subsets can gain access to the secret?
Well, the easiest way I can see to represent this access structure is:


Select three random values $r_1, r_2, r_3$
Generate a threshold-2 secret sharing scheme with the secret $S$ (where $S$ is the ultimate secret), and whose shares are $\alpha_1, \alpha_2, \alpha_3$
Generate an independent threshold-2 secret sharing scheme with the secret $r_3$, and whose shares are $\beta_1, \beta_2, \beta_3$.


Then,


$A$ gets $\alpha_1$, $r_3 \oplus S$
$B$ gets $r_1$, $\beta_1$
$C$ gets $r_2$, $\beta_2$
$D$ gets $r_1 \oplus r_2 \oplus S$, $\beta_3$
$E$ gets $\alpha_2$, $r_3 \oplus S$
$F$ gets $\alpha_3$, $r_3 \oplus S$


It's easy to see that any of the allowed subsets can recover $S$ (e.g. for the subset $A, B, C$, they jointly know $\beta_1$ and $\beta_2$, that allows them to recover $r_3$; and with $r_3 \oplus S$ gives them $S$).  It should be fairly clear that any subset not within the access structure gets no information about $S$.

This doesn't give an answer to the precise question to raised; it does appear to solve the problem you were trying to address.