Sunflowers in Set Systems of Bounded Dimension

Given a family F of k-element sets, S1, . . . , Sr ∈ F form an r-sunflower if Si ∩ Sj = Si′ ∩ Sj′ for all i ̸= j and i′ ̸= j′. According to a famous conjecture of Erdős and Rado (1960), there is a constant c = c(r) such that if |F| ≥ c, then F contains an r-sunflower. We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, VC-dim(F) ≤ d. In this case, we show that r-sunflowers exist under the slightly stronger assumption |F| ≥ 210k(dr) 2 log∗ k . Here, log∗ denotes the iterated logarithm function. We also verify the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems. 2012 ACM Subject Classification Mathematics of computing → Combinatoric problems

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