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

Abstract

Let f(k, n, Σ denote the maximum of ∥$\text{F}$∥, 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 Bollobás 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.