Results on ordered copies, anchor rigidity and distance from freeness, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Order relations of the tree positions — Lemma 3.1. Child and parent blocks compare by the parent index: R_i^+(p) < R_{i-1}^+(a) iff pi(p) <= a, and three analogous equivalences.
• Properties of the anchor — Lemma 1.1. The anchor S has distinct rows and columns, dense first 32 rows and columns, and all five-bit words in its corner.
• No five non-anchor columns are shattered — Lemma 1.3. Non-anchor rows realize at most 27 of the 32 words on any five non-anchor columns.
• Anchor rigidity (OpenAI) — Lemma 2.1. Every ordered copy of S in A_h uses one position of each anchor group of a single mode, in order.
• Copies meet the leaf diagonal (OpenAI) — Proposition 3.1. Every ordered copy of H in A_h passes through one of the 2^h diagonal leaf cells, so N_H(A_h) <= 2^h n^130.
• Distance from H-freeness (OpenAI) — Proposition 1.1. Every H-free binary matrix differs from A_h in at least 4^h cells.
• Polynomial ordered removal fails (OpenAI) — Theorem 2.1. A_h is (386h+2)^-2-far from H-free and has at most (386h+2)^-2 2^-h n^132 ordered copies of H.
• Canonical row-column sampling — Corollary 2.3. Sampling q rows and q columns finds H in A_h with probability at most eps_h 2^-h q^2.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.