I’ve changed my topic plan slightly. Please open the index page. We’ll finish the review of Paper S4 in the next post. To fully convince ourselves of the main claim of Paper 3, we’ll discuss the two topics other than the rank-g construction as mentioned in the 11/13/23 post: 2. the push-up lemma, and 3. initial core growth. The third topic is about splitting sets and families involving Lemma A.1 of Paper S4 and the split lemma given in the 3/3/23 post, appropriate for the last topic of the current chapter.
Then I’ll explain topic 2 and main recursive steps in the next chapter to finish presenting all the materials in Paper 3. I’ll publish weekly again during summer.
Back to Step 2 of the main construction for Proposition 2.1, please recall the terminology defined last time. We confirm the \( \Gamma_g(b_*, h) \)-condition in the rest of this post.
Note about C) that \[ m_*= \frac{m n_*}{n}, \quad b>m^{1+\epsilon} \frac{n_*}{n}=m_* m^{\epsilon}, \quad \textrm{and} \quad \frac{1}{g} < \epsilon^2, \] by definition. So \( b_\dagger= \frac{b^{1-1/g}}{m^{1/g}} \) is greater than \( m_* m^{\epsilon/2} \), which makes sense for us to consider the \( \Gamma( b_\dagger, h_T ) \)-condition there. Also \( b_* = \frac{b_\dagger}{2^g (g-1) m_*} \) in D) is greater than any constant because so is m.
Next time, we’ll apply the obtained \( \Gamma_g\)-condition to the \( g^{th} \)-mark lemma to confirm the desired exported collective \( \Gamma \)-condition.
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