A family of r distinct sets \(\{A_1,\ldots , A_r\}\) is an r-sunflower if for all \(1 \leqslant i < j\leqslant r\) and \(1 \leqslant i' < j'\leqslant r\) , we have \(A_i\cap A_j = A_{i'}\cap A_{j'}\) . Erdős and Rado conjectured in 1960 that every family \(\mathcal {H}\) of \(\ell \) -element sets of size at least \(K(r)^\ell \) contains an r-sunflower, where K(r) is some function that depends only on r. We prove that if \(\mathcal {H}\) is a family of \(\ell \) -element sets of VC-dimension at most d and \(|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell \) for some absolute constant \(C > 0\) , then \(\mathcal {H}\) contains an r-sunflower. This improves a recent result of Fox, Pach, and Suk. When \(d=1\) , we obtain a sharp bound, namely that \(|\mathcal H| > (r-1)^\ell \) is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.
