New cases of Reay’s conjecture on partitions of points into simplices with -dimensional intersection

Reay’s conjecture asserts that every set of (m−1)(d+1)+k+1 points in general position in (with 0≤k≤d) has a partition X₁,X₂,…,X_m such that is at least k-dimensional. We prove this conjecture in several cases: when m≤8 (for arbitrary d and k), when d=6,d=7 and d=8 (for arbitrary m and k), and when k=1 and d≤24 (for arbitrary m).