On a d-degree Erd{o}s-Ko-Rado Theorem

A family of subsets F is intersecting if A ∩ B ≠ ∅ for any A, B ∈ F. In this paper, we show that for given integers k > d ≥ 2 and n ≥ 2k+2d-3, and any intersecting family F of k-subsets of \1, ⋯, n\, there exists a d-subset of $[n]$ contained in at most n-d-1k-d-1 subsets of F. This result, proved using spectral graph theory, gives a d-degree generalization of the celebrated Erd{o}s-Ko-Rado Theorem, improving a theorem of Kupavskii.