For d = k-1 the n-threshold improves to 2k + √(2k) + O(1); for d = k-2 it improves to 2k + 7k^{2/3} + O(k^{1/3}).
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Note on the codegree version of the Erd\H{o}s--Ko--Rado theorem
For d = k-1 the n-threshold improves to 2k + √(2k) + O(1); for d = k-2 it improves to 2k + 7k^{2/3} + O(k^{1/3}).