The Ramsey number of loose cycles versus cliques

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Recently Kostochka, Mubayi and Verstraëte initiated the study of the Ramsey numbers of uniform loose cycles versus cliques. In particular they proved that R(Cʳ₃,Kʳₙ) = θ(n³/²) for all fixed r≥ 3. For the case of loose cycles of length five they proved that R(C₅ʳ,Kₙʳ)=Ω((n/ n)⁵/⁴) and conjectured that R(Cʳ₅,Kₙʳ) = O(n⁵/⁴) for all fixed r≥ 3. Our main result is that R(C₅³,Kₙ³) = O(n⁴/³) and more generally for any fixed l≥ 3 that R(Cₗ³,Kₙ³) = O(n1 + 1/(l+1)/2). We also explain why for every fixed l≥ 5, r≥ 4, R(Cʳₗ,Kʳₙ) = O(n1+1/ l/2) if l is odd, which improves upon the result of Collier-Cartaino, Graber and Jiang who proved that for every fixed r≥ 3, l≥ 4, we have R(Cₗʳ,Kₙʳ) = O(n1 + 1/(l/2 -1)).