Hegedus Gabor: New upper bounds for the size of a sunflower-free family.

We say that a subset $A \subseteq \{1, . . . , D\}^n$
for D > 2 is sunflower-free if for every
distinct triple $x, y, z \in A$ there exists a coordinate i where exactly two of
$x_i, y_i, z_i$ are equal. Using the polynomial method with characters Naslund
and Sawin proved that any sunflower-free set $A\subseteq \{1, . . . , D\}^n$ has size

$|$A| \le c^n_D$$,

where $c_D =\frac{3}{2^{2/3}} (D − 1)^{2/3}.
.

In this lecture I give a new upper bound for the size of $k$-uniform sunflower-
free set families. In the proof I use Naslund and Sawin’s result about

sunflower-free subsets $A \subseteq \{1, . . . , D\}^n$.