Sunflower-Free Uniform Families: Recursive Constructions and Explicit Bounds

Abstract

Let $f(w,k)$ be the maximum size of a $w$-uniform family containing no sunflower with $k$ petals. We introduce a recursive construction for sunflower-free families and use it to obtain a general lower bound on the exponential growth rate of $f(w,k)$. We also prove a general upper bound for $3$-uniform families with at least four petals. Our results give $39\le f(3,4)\le 49$, $f(3,5)\le 146$, $153\le f(3,6)\le 255$, $259\le f(3,7)\le 474$, and $54\le f(4,3)\le 83$. In addition, we prove that the maximum size of an intersecting $4$-uniform family containing no sunflower with three petals is $27$. The upper bounds $49$ and $83$ are computer-assisted. The finite lower bounds come from explicit constructions.