On higher real $K$-theories and finite spectra

We study higher chromatic height analogues $e{o}_h$ of the connective real $K$-theory spectrum $ko$. We show that $e{o}_h$ is an fp spectrum of type $h$ in the sense of Mahowald–Rezk. We use these to study an Euler characteristic for fp spectra introduced by Ishan Levy, and give a partial answer to a question of Levy regarding the algebraic $K$-theory of the category of finite type $h$ spectra. As a corollary, we prove that if the generalized Moore spectrum $\mathbb{S}\mathrm{/}(2^{i_0},v_1^{i_1},\dots ,v_h^{i_h})$ exists, then the $2$-adic valuation of $\prod i_j$ must exceed that of the height $h$.