Sunflowers in set systems of bounded dimension (Q6057492)

Given a family \(\mathcal{F}\) of \(k\)-element sets, \(S_1,\ldots ,S_r \in \mathcal{F}\) form an \(r\)-sunflower if \(S_i \cap S_j =S_{i'} \cap S_{j'}\) for all \(i \neq j\) and \(i'\neq j'\). A famous conjecture of \textit{P. Erdős} and \textit{R. Rado} [J. Lond. Math. Soc. 35, 85--90 (1960; Zbl 0103.27901)] asserts that there is a constant \(c = c(r)\) such that if \(\vert \mathcal{F}\vert \geq c^k \), then \(\mathcal{F}\) contains an \(r\)-sunflower. In this paper, the authors prove this conjecture for families of bounded Vapnik-Chervonenkis dimension, VC-dim(\(\mathcal{F})\leq d\) and in this case, they show that \(r\)-sunflowers exist under the slightly stronger assumption \(\mathcal{F}\vert \geq 2^{10k(dr)^{2 \log^\ast k}}\). Here, log\(^\ast\) denotes the iterated logarithm function. They also verify the Erdős-Rado conjecture for families \(\mathcal{F}\) of bounded Littlestone dimension and for some geometrically defined set systems. One conjecture concludes the paper.