Subquadratic bound for higher interval chromatic number
Determine whether, for fixed positive integers d and χ, there exists ε=ε(d,χ)>0 such that every n-vertex ordered 3-uniform hypergraph H^< with maximum degree d and interval chromatic number χ satisfies R_<(H^<)≤2^{O(n^{2-ε})}.
References
Another interesting problem is to extend the upper bound with subquadratic exponent from Theorem~\ref{thm-ordRamHyper-3UnifMaxDegIntChr} to ordered $3$-uniform hypergraphs with bounded maximum degree and fixed interval chromatic number that is larger than $3$.
Let $d$ and $\chi$ be fixed positive integers. Is there an $\varepsilon = \varepsilon(d,\chi)>0$ such that, for every ordered $3$-uniform hypergraph $H<$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $\chi$, we have \R_<(H<) \leq 2{O(n{2-\varepsilon})}?\
Is there an $\varepsilon = \varepsilon(d,\chi)>0$ such that, for every ordered $3$-uniform hypergraph $H<$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $\chi$, we have R_<(H<) \leq 2{O(n{2-\varepsilon})}?