Having finished our proof of Lemma A.1 of Paper S4 last time, the bound (4) given on 6/17/24 is now available to help resolve our initial core growth problem. First, let’s find a general good \( r\)-split following the same style as Theorem 4 on 6/3/24.
To prove Theorem 5 from (5), we again count the pairs \( (U, \boldsymbol{X} )\) using the same technique as 5/13/24, 11/6/23 and the extra notes of the 2/23 post: among the \( |{\mathcal F}| \prod_{i=1}^r {n-(i-1)n/r \choose n/r} \) such pairs, less than \( m \exp \left( – \frac{m}{12 r} \right) \) of them fail to meet \( \frac{m}{2r} < |U \cap X_1| < \frac{2m}{r} \) due to (5); count them the same way as 5/13/24 fixing each \( U \). By symmetry, less than \( m \exp \left( – \frac{m}{12 r} \right) \) of them fail to meet \( \frac{m}{2r} < |U \cap X_i| < \frac{2m}{r} \) for any particular \( X_i \).
So, all the other \( (U, \boldsymbol{X} )\) that amount to \( \left[ 1- r m \exp \left( – \frac{m}{12 r} \right) \right] |{\mathcal F}| \prod_{i=1}^r {n-(i-1)n/r \choose n/r} \) or more meet the double inequality for every strip \( X_i \). Hence, there must be an \( \boldsymbol{X} \) and subfamily \( {\mathcal F’ } \subset {\mathcal F} \) meeting the two conditions given by Theorem 5, completing its proof.
The theorem helps us justify the simultaneous core growth presented on 6/17/24 for \( k=3 \) completely. Given an \( {\mathcal F} \) satisfying \( \Gamma (b) \) where \( b=m^{\frac{1}{2} + \epsilon} \), find a good \( \boldsymbol{X} \) and \( {\mathcal F}’ \) by the theorem. For \( j=1, 2, \ldots, r \), we recursively prove the existence of:
- mutually disjoint three sets \( C_i \) with \( |C_i | = O( j r \ln m ) \),
- and subfamilies \( {\mathcal F}_i \subset {\mathcal F}'[C_i] \) for \( i \in [3] \),
- satisfying the \( \Gamma (b_j, 2) \)-condition in \( X – C_i \), i.e., \( | {\mathcal F}_i [S] | < 2 b_j^{-|S|} | {\mathcal F}_i| \) for every nonempty \( S \subset X – C_i \), where \( b_j = b( 1- 1/r)^j \),
- and \( |U \cap X_{j’}| = q_{j’} \) for each \( j’ \le j \) and \( i \in [3]\), some \( q_{j’} \in \left( \frac{m}{2r}, \frac{2m}{r} \right) \), and all \( U \in {\mathcal F}_i \).
You can check the truth for each \( j \) by performing the process described there. Next time, we’ll generalize this to any given \( k \) to pose our final solution to the initial core growth problem.
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