Extremal problems whose solutions are the blowups of the small witt-designs

Abstract Let f(k, n, Σ denote the maximum of ∥ F ∥, where F⊂({1,…,n}k) and there are no F1, F2, F3 ϵ F with ∥F1 ∩ F2∥ = k − 1, F1ΔF2 ⊂ F3. The function f(k, n, Σ) was determined for k = 2 by Mantel, for k = 3 by Bollobas and for k = 4 by Sidorenko. Here we determine it for k = 5, 6 and n > n0. Moreover,we show that the only optimal families, i.e., ∥F∥ = f(k, n, Σ) arise from the unique (11, 5, 4) or (12, 6, 5) Steiner-systems by a simple operation, called blowup.