We are in the last step to prove Proposition 2.1 of Paper S4. Step 3 on p. 9-10 derives the desired exported collective \( \Gamma \)-condition of most \( {\mathcal F}_Y \), which we’ll confirm in this post as well. Please recall that in the 11/23/24 post and also in Step 1, we checked the truth of the density requirement \( |{\mathcal F}_Y | > \frac{1}{2} {l \choose m_*} {n_* \choose m_*}^{-1} |{\mathcal F}| \) for more than \( (1-\epsilon^2) { n_* \choose \epsilon n_*} \) sets \( Y \in {X_* \choose l} \).
So we can apply the \( g^{th}\) mark lemma to \( {\mathcal H} \) with the \( \Gamma_g \)-condition. Since \( \sqrt c \) is much larger than \( \epsilon^{-2} \):
With both i) and ii) on the bottom of p. 7 confirmed, we’ve now finished our proof of Proposition 2.1. By this, we demonstrated that the collective \( \Gamma \)-condition can be exported to most \( {\mathcal F}_Y \). The method based on the rank-g construction can be effective to prove Proposition 3 given in the 11/13 post. Its confirmation is our current goal that implies \( \Gamma (m^{\frac{1}{2} + \epsilon} ) \) of \( {\mathcal F} \) \( \Rightarrow \) three disjoint sets in \( {\mathcal F} \).
We’ll start our next topic next time.
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