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Home » Title 45 Ends in Disaster—Dive Into Title 46 Now!, - SUpost

Title 45 Ends in Disaster—Dive Into Title 46 Now!, - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 Suppose the 3 edges in \( E \) have intersection structure:

📌 Case 1: The three close pairs share a common vertex â e.g., a star of 3 edges: \( AB, AC, AD \). But we only have 3 edges â this uses all 3 close pairs incident to A. Then consider trio \( B,C,D \): no edges among them (unless present), so no close pairs â fails. Trio \( A,B,C \): only \( AB \) is close â only 1 close pair â fails.

📌 So star with 3 edges: only 3 close pairs, but no trio has exactly 2.

📌 Case 2: The three close pairs form a path: \( AB, BC, CD \). Edges: AB, BC, CD. Trio \( A,B,C \): AB, BC close â 2 close pairs. Trio \( B,C,D \): BC, CD close â 2 close pairs. Trio \( A,B,D \): AB, BD? BD not close, AD not close â only AB â 1 close â fails. Trio \( A,C,D \): AC not close â 0 â fails. So only the two middle trios (A,B,C) and (B,C,D) each have exactly 2 close pairs.

📌 So one two-edge path with distinct middle vertex.

📌 Number of such paths: a path of 3 nodes has 2 endpoints and one middle. Number of such paths in a complete graph of 5 nodes: number of ways to choose 3 distinct nodes: \( inom{5}{3} = 10 \), each induces a path unless it's complete or linear.

📌 But we fix the set of 3 close pairs. We need among all 3-edge sets with exactly two edges forming a path of length 2, how many such sets exist.

📌 Alternate approach: total number of trios with exactly two close pairs.

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