图论竞赛图问题：请求排查3个证明的逻辑瑕疵

Hey folks! I’ve wrapped up proofs for three tournament graph problems in graph theory, and I’m hoping to get some fresh eyes from the community to catch any possible logical flaws or gaps in my reasoning. Let me start by laying out the key definitions we’re working with to set the stage:

• n-Tournament Graph: A complete directed graph with n "players" as vertices, where for any two distinct players x and y, exactly one of the asymmetric relations holds: x beats y (written as x → y) or y beats x—no ties exist.
• Player Score: The total number of other players that a given player has beaten.
• Alpha Player: A player k is classified as an alpha player if and only if for every other player r (r ≠ k), one of the following is true:
  1. k directly beats r (k → r), OR
  2. There exists some intermediate player s such that k beats s, and s beats r (forming a path k → s → r).

I’ve formalized proofs for three distinct problems centered on these definitions. If you’re willing to help review, I can share the full proof text for each problem individually—just let me know which one you’d like to dive into first! I’m particularly keen on checking for:

• Overlooked edge cases (like small n values, e.g., n=2, n=3)
• Unstated assumptions that aren’t actually implied by the definitions
• Logical leaps where a step might seem intuitive but needs explicit justification
• Incorrect applications of tournament graph properties (like score sequence properties or transitivity subsets)

内容的提问来源于stack exchange，提问作者rachelhoward