The final four manuscripts in OpenAI’s release address longstanding geometric and combinatorial questions spanning discrete lattices, Ramsey growth rates, and graph extremal numbers.

1. Settling Ehrhart's Volume Conjecture

For any full-dimensional convex body (K \subset \mathbb{R}^n) whose barycenter is the origin and whose only interior integer point is the origin, the paper proves the sharp upper bound:

vol⁑(K)≀(n+1)nn!\operatorname{vol}(K) \le \frac{(n+1)^n}{n!}

The breakthrough translates the discrete lattice problem into complex analysis on ((\mathbb{C}^*)^n), using Bergman-kernel positivity on weighted holomorphic function spaces.

2. Multicolor Triangle Ramsey Numbers: (R_k(3) = k^{\Theta(k)})

By recursively gluing complete graphs with coordinate-cover matrices that prevent monochromatic triangles, the paper proves a superexponential lower bound, finally establishing the exact asymptotic growth class:

Rk(3)=kΘ(k)R_k(3) = k^{\Theta(k)}

πŸ“š

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