Diameters of uniform subset graphs

Let r,sr,s be positive integers with r>sr>s, kk a nonnegative integer, and n=2r−s+kn=2r−s+k. A uniform subset graph G(n,r,s)G(n,r,s) is a graph with vertex set [n]r[n]r and where two rr-subsets A,B∈[n]rA,B∈[n]r are adjacent if and only if |A∩B|=s|A∩B|=s. Let diam(G) denote the diameter of a graph GG.In this paper, we prove the following results: (1) If k>0k>0, then diam(G(n,r,s))=⌈r−s−1s+k⌉+1 if r≥2s+k+2r≥2s+k+2, 2 if k≥sk≥s and 2s≤r≤s+k2s≤r≤s+k, or k