Sunflowers in Set Systems of Bounded Dimension

Given a family $${\mathcal {F}}$$ of k-element sets, $$S_1,\ldots ,S_r\in {\mathcal form an r-sunflower if $$S_i \cap S_j =S_{i'} S_{j'}$$ for all $$i \ne j$$ and $$i' j'$$ . According to famous conjecture Erdős Rado (JAMA 35: 85–90, 1960), there is constant $$c=c(r)$$ such that $$|{\mathcal {F}}|\ge c^k$$ , then contains r-sunflower. We come close proving this families bounded Vapnik-Chervonenkis dimension, $$\text {VC-dim}({\mathcal {F}})\le d$$ In case, we show r-sunflowers exist under the slightly stronger assumption 2^{10k(dr)^{2\log ^{*} k}}$$ Here, $$\log ^*$$ denotes iterated logarithm function. also verify Erdős-Rado Littlestone dimension some geometrically defined set systems.