With Theorem 5 proven last time, we can now present a general statement to resolve our initial core growth problem.
This surely offers a solution to the problem: by Proposition 3 on 11/13/23, we are given an \( {\mathcal F} \) with the \( \Gamma ( cm^{\epsilon + \frac{1}{2}} k \ln^2 k ) \) condition. The theorem provides the three objects we need to perform the rank-\(g\) construction on \(X_1 \) and other \(X_j \).
To prove it, use Theorem 5 to obtain an \( r \)-split \( \boldsymbol{X} \) and subfamily \({\mathcal F’} \). Then recursively confirm the following proposition by the simultaneous core growth.
The approximate equality is due to \( r \gg 1 \) \( \Rightarrow \) \( (1- r^{-1})^{-r} \simeq e \), simpler than (1) on 7/20/23 to see. Perform the updates \[ \textrm{(7)} \qquad C_i \leftarrow C_i \cup S, \qquad \textrm{and} \qquad {\mathcal F}_i \leftarrow {\mathcal F}’_i [S], \] for \( i=1 \). Then \( C_1 \) and \( {\mathcal F}_1 \) meet the four required conditions of \( \Psi(j) \).
We’ll continue this to finish the proof of Theorem 6 next time. Note \( \Psi(r) \) implies the theorem.
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