The Exact Erdős–Ko–Rado Theorem for 3-Wise 𝑡-Intersecting Uniform Families

One of the central problems in extremal set theory is the determination of the maximum size of an 𝑟-wise 𝑡-intersecting 𝑘-graph on 𝑛 vertices. For 𝑟 = 2 after decades of effort, a complete solution was given by the Complete Intersection Theorem ([2]). However the case 𝑟 ≥ 3 remains wide open. The 𝑡-star consisting of all 𝑘-sets containing a fixed 𝑡-set provides a lower bound. For 𝑟 = 3 and 𝑘 ≥ 𝑡 + 2 ≥ 44, we determine the exact threshold for which it is optimal for 𝑛 > 𝑘. The condition is essentially 𝑛 ≥ √𝑡𝑘. For 𝑘 ≥ 𝑡 + 2 ≥ 47 and 𝑛 ≥ √4𝑡+9−1 2 𝑘, we also establish the corresponding Hilton–Milner-type stability result.