Does large chromatic or cochromatic number force large dichromatic number? (Erdős #761)

The cochromatic number of a graph $G$, denoted $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ so that each colour class induces either a complete graph or an empty graph. The dichromatic number of $G$, denoted $\delta(G)$, is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$, there is a $k$-colouring of the vertices with no monochromatic oriented cycle. Must a graph with large chromatic number have large dichromatic number — i.e. does $\chi(G)\to\infty$ force $\delta(G)\to\infty$? Must a graph with large cochromatic number contain a subgraph with large dichromatic number?

Acceptance. FULLY RESOLVES: a proof that for every $k$ there is $r(k)$ such that $\chi(G)\geq r(k)$ implies $\delta(G)\geq k$ (first question), or of the stronger cochromatic statement (second question); OR a refutation — an explicit infinite family with unbounded chromatic (resp. cochromatic) number and uniformly bounded dichromatic number, with full proofs of both properties. Machine-checkable (Lean/Coq) proof preferred, else a complete written proof. ADVANCES: (a) exhibit a finite $c$ such that $\chi(G)\geq c$ forces $\delta(G)\geq 3$ (the first nontrivial threshold); (b) prove either implication for a natural restricted class (e.g. bounded clique number, Kneser-type families); (c) verified computations of dichromatic numbers of concrete structured families that materially map the frontier, with code and a per-graph certificate (an orientation plus an exhaustive or SAT-certified check); (d) a proof clarifying the second question's relation to the first beyond the known one-way implication. Deliver the proof file, or the counterexample family with proofs, or the computation code plus certificates.