Kostochka–Mubayi–Verstraëte loose-triangle Ramsey conjecture

For fixed $k\geq3$, let $LC_3^{(k)}$ be the $k$-uniform loose 3-cycle, whose consecutive edges intersect in exactly one vertex and whose nonconsecutive edges are disjoint. Write $r_k(LC_3,n)$ for the Ramsey number of $LC_3^{(k)}$ versus the complete $k$-uniform hypergraph on $n$ vertices. Kostochka–Mubayi–Verstraëte conjecture.

$$r_k(LC_3,n)=o(n^{3/2}).$$

The known bounds have order between $n^{3/2}/(\log n)^{3/4}$ and $n^{3/2}$ for $k=3$, with $n^{3/2+o(1)}$ known for all $k\geq3$. The conjecture seeks the hypergraph analogue of the logarithmic improvement for graph triangles.