After proving Proposition 2.1 of Paper S4 for g=2, we found a way to generalize the method to a larger rank g last time. On a g-parameter weight \( w \) with the associated norm \( \| \cdot \| \) and sparsity \( \kappa (\cdot ) \), we want to consider \( g^{th} \) marks \( ( {\boldsymbol U}, Y) \), the neighbor tuple families \( {\mathcal P}_{j, g} \), and the new \( \Gamma_g(b, h) \)-condition defined on p. 1-2. These will support our method of the rank-g construction for proving Prop. 2.1 completely.
Among the four tasks presented, we’ve finished the first one with the defined \( \Gamma_g(b, h) \)-condition that mainly cares about the union of the \( U_i \) in \( {\boldsymbol U} \) for manageable proof complexity. The notation \( Union ({\boldsymbol U} ) = \bigcup_{i=1}^g U_i \) used there is something we should keep in mind afterward. The family \( {\mathcal P}_{j, g} \) collects \( {\boldsymbol U} = (U_1, U_2, \ldots, U_g) \in {\mathcal F}^g \) such that \( Union({\boldsymbol U}) = gm-j \), consistently with the case g=2. The \( \Gamma_2 \)-condition requires \( \| {\mathcal P}_{j, g} \| < h b^{-j} \| {\mathcal F} \|^g \) for \( j \in [m] \). Its RHS has no binomial coefficient such as \( {(g-1)m \choose j} \), instead of which the considered b value is \( 4 \gamma n / l \) unlike \( b= 5 \gamma n m / l \) so far.
Let’s move to the second task to generalize the double mark lemma. We want to show that the \( \Gamma_g \)-condition means $$ \| {\mathcal D}_g \| < \frac{\left( 1+ \frac{h}{\gamma} \right){n \choose l}{l \choose m}^g}{{n \choose m}^g} \| {\mathcal F} \|^g = \left( 1+ \frac{h}{\gamma} \right){n \choose l}{l \choose m}^g e^{-g \cdot \kappa ({\mathcal F})}, $$ where \( {\mathcal D}_g \) is the family of \( g^{th} \) marks \( ( {\boldsymbol U}, Y) \), and its norm is the sum of \( w( {\boldsymbol U} ) \) for all \( ( {\boldsymbol U}, Y) \). Lemma 1.1 on p. 2 states this.
We may drop the subscript g when its value is clear, so we can write \( {\mathcal D} \), \( {\mathcal P}_j \), etc. just as before. Below the description of the \( g^{th} \) mark lemma, the paper says:
Those are common with g=2 being clear except for \( j \le (g-1)m \): the subscript j of \( {\mathcal P}_{j, g} \) comes from \( Union({\boldsymbol U}) = gm-j \), so j must not exceed \( (g-1) m\).
Since \( \| {\mathcal D} \| \) is the sum of \( \left\| {Y \choose l} \right\|^g =w \left( {\mathcal F}_Y \right) \) for all l-sets Y, the \( g^{th} \) mark lemma implies that most Y meet \( w \left( {\mathcal F}_Y \right) < \epsilon^{-2} \left( 1+ h \gamma^{-1} \right) {l \choose m}^g {n \choose m}^{-g} \| {\mathcal F} \|^g \).
By the weighted EGT, most Y satisfy the density requirement \( w( |{\mathcal F}_Y ) > \frac{1}{2} {l \choose m_*} {n_* \choose m_*}^{-1} |{\mathcal F}| \) with \( n \rightarrow n_* \) and \( m \rightarrow m_* \) as the 11/27 post. So if the g-parameter weight \( w \) is set as Step 2 on p. 8, we will have most Y to meet both the density requirement and exported \(g^{th} \) collective \( \Gamma \)-condition. That’s what Step 3 concludes on p. 10 finishing the proof of Prop. 2.1.
We’ll prove the \( g^{th} \) mark lemma next time.
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