A recursive coloring function without ${\mathrm{\Pi }}_3^0$ solution for Hindman theorem

We show that there exists a recursive coloring function $c$ such that any ${\mathrm{\Pi }}_3^0$ set is not a solution to $c$ for Hindman’s theorem. We also show that there exists a recursive coloring function $c$ such that any ${\mathrm{\Delta }}_3^0$ set is not a solution to $c$ for Hindman’s theorem restricted on sum of at most three numbers.