Off-diagonal Ramsey numbers for linear hypergraphs

We study off-diagonal Ramsey numbers 𝑟⁡(𝐻,𝐾(𝑘) 𝑛) of 𝑘-uniform hypergraphs, where 𝐻 is a fixed linear 𝑘-uniform hypergraph and 𝐾(𝑘) 𝑛 is complete on 𝑛 vertices. Recently, Conlon, Fox, Gunby, He, Mubayi, Suk, and Verstraëte disproved the folklore conjecture that 𝑟⁡(𝐻,𝐾(3) 𝑛) always grows polynomially in 𝑛. In this paper, we show that much larger growth rates are possible in higher uniformity. In uniformity 𝑘 ≥4, we prove that for any constant 𝐶 >0, there exists a linear 𝑘-uniform hypergraph 𝐻 for which 𝑟⁡(𝐻,𝐾(𝑘) 𝑛)≥twr𝑘−2⁢(2(log⁡𝑛)𝐶).