The answer to the first question is no. Suppose $|M|=11$ and $\mathcal S\subset\binom M5$ and $|\mathcal S|=77.$ Since each $5$-set contains five $4$-sets, and since $77\cdot5=385\gt330=\binom{11}4,$ by the pigeonhole principle some $4$-set must be contained in two members of $\mathcal S,$ which are both contained in the same $6$-set.
bof
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