In this post, we start reviewing the new paper S4 to eventually prove its Proposition 2.1 we saw last time. Given an \( n_* \)-set \(X_* \) with \( m_* = m n_* / n \) in addition to \( {\mathcal F} \subset {X \choose m} \), our main motivation of the rank-g construction is not just to show there are a majority of \( Y \in {X_* \choose \epsilon n_*} \) such that \( {\mathcal F}_Y = {\mathcal F} \cap {X – X_* \cup Y \choose m} \) are reasonably dense (as in the 10/15 post and 11/19 post), but also to have each \( {\mathcal F}_Y \) satisfy the \(g^{th} \) collective \( \Gamma \)-condition. Let’s give it an initial thought about how we could achieve this feature we called export of the \( \Gamma \)-condition to \( X – X_*\).
Here we assume that every m-set \( U \in {\mathcal F} \) intersects with \(X_* \) by the cardinality \( m_* \). This can be safely presumed with the method 3 mentioned last time using Lemma A.1 on p. 10-12 of Paper S4. Paper S3 achieves this with a generalized split lemma also, but there is a simpler way. Paper S4 briefly mentions about it in the 4th paragraph of p. 7. We’ll discuss this in a later post.
With g=2 for simplicity, we look for a weight function w as follows:
Such a \( w \) can’t be defined on \( 2^{X_*} \). We can try \( w(U) = \alpha |{\mathcal F}[U]| \) and \( w(U) = \alpha |{\mathcal F}[U]|^2 \) for some coefficients \( \alpha>0 \) or others to see none of them works: it’s hard to correctly raise \( |{\mathcal F}_Y[T]| \) to the second power on the RHS of \( w ({\mathcal F}_Y ) \) with just one parameter of \( w: 2^{X_*} \rightarrow {\mathbb R}_{\ge 0} \).
However, it works if w is defined by \[ (19) \qquad w: 2^{X_*} \times 2^{X_*}\rightarrow {\mathbb R}_{\ge 0}, \qquad (U_1, U_2) \mapsto \sum_{T \subset X – X_*,~T \ne \emptyset} \frac{|{\mathcal F}[U_1 \cup T]| ~~ |{\mathcal F}[U_2 \cup T]|}{b^{-|T|} {m \choose |T|}}. \] Because with this two-parameter \(w \), each \( w ( {\mathcal F}_Y ) \) equals \[ \sum_{(U_1, U_2) \in {Y \choose m_*}^2} w(U_1, U_2) = \sum_{(U_1, U_2) \in {Y \choose m_*}^2} ~~ \sum_{T \subset X – X_,~T \ne \emptyset} \frac{|{\mathcal F}[U_1 \cup T]| ~ |{\mathcal F}[U_2 \cup T]|}{b^{-|T|} {m \choose |T|}} \] \[ = \sum_{T \subset X – X_,~T \ne \emptyset} ~~ \frac{|{\mathcal F}_Y[T]|^2}{b^{-|T|} {m \choose |T|}}, \] as required in the figure. Here \( {Y \choose m_*}^2 \) is a shorthand for \( {Y \choose m_*} \times {Y \choose m_*} \) working for other values of g and families the same way. The above is true because:
- \( |{\mathcal F}_Y[T]|^2 \) in the second line counts the number of pairs \( (V_1, V_2) \in {\mathcal F}_Y[T]^2 \).
- As assumed, it’s enough to consider all the \( m_* \)-subsets U of \( X_* \), and \( {\mathcal F}[U] \) that are mutually disjoint.
- So if you sum up \( |{\mathcal F}[U_1 \cup T]| ~ |{\mathcal F}[U_2 \cup T]| \) for each T and all \( (U_1, U_2) \in {Y \choose m_*}^2 \), you get \( |{\mathcal F}_Y[T]|^2 \), since every \( (V_1, V_2) \) is correctly counted just once in both.
Therefore, we need a g-parameter weight \( w: (2^{X_*})^g \rightarrow {\mathbb R}_{\ge 0} \) to be able to export a \(g^{th} \) collective \( \Gamma \)-condition. That’s why we want the rank g as the first page of Paper S4 defines.
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