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Given a family $\mathcal F$ of $k$-element sets, $S_1,\ldots,S_r\in\mathcalF$ form an {\em $r$-sunflower} if $S_i \cap S_j =S_{i'} \cap S_{j'}$ for all $i\neq j$ and $i' \neq j'$. According to a famous conjecture of Erd\H os and Rado(1960), there is a constant $c=c(r)$ such that if $|\mathcal F|\ge c^k$, then$\mathcal F$ contains an $r$-sunflower.  We come close to proving this conjecture for families of bounded {\emVapnik-Chervonenkis dimension}, VC-dim$(\mathcal F)\le d$. In this case, weshow that $r$-sunflowers exist under the slightly stronger assumption$|\mathcal F|\ge2^{10k(dr)^{2\log^{*} k}}$. Here, $\log^*$ denotes the iteratedlogarithm function.  We also verify the Erd\H os-Rado conjecture for families $\mathcal F$ ofbounded {\em Littlestone dimension} and for some geometrically defined setsystems.

Sunflowers in set systems of bounded dimension