# Lemma 7

## Lemma 7:

This proof is much longer than anticipated (and it's still not completed), so it will be split into six cases (with each case relying on the previous cases):

Lemma 7.1: If there are two size 5 sets intersecting at 4 elements, then $\mathcal{A}$ is Frankl's

Lemma 7.2: If there are two size 5 sets intersecting at 3 elements, then $\mathcal{A}$ is Frankl's

Lemma 7.3: If there are two size 5 sets intersecting at 2 elements, then $\mathcal{A}$ is Frankl's

Lemma 7.4: If there are two intersecting size 5 sets, then $\mathcal{A}$ is Frankl's

Lemma 7.5: If there are two size 5 sets, then $\mathcal{A}$ is Frankl's

Lemma 7.6: If there is a size 5 set, then $\mathcal{A}$ is Frankl's