On cardinal invariant concerning the family of Hamel functions (Q1772987)

This is an abstract of the lecture during the Summer Symposium in Real Analysis XXVIII, Slippery Rock, PA, June 8--13. The author shows that for the family HR of all Hamel functions the equality A(HR)=\(\omega\) holds. Recall that \(f\colon \mathbb{R} \to\mathbb{R}\) is a Hamel function if \(f\) considered as a subset of \(\mathbb{R}^2\) is a Hamel basis for \(\mathbb{R}^2\). For \({\mathcal F}\subset\mathbb{R}^\mathbb{R}\) A(\(\mathcal F\)) denotes the size of the smallest family \(G\subset\mathbb{R}^\mathbb{R}\) such that there is no \(g\in\mathbb{R}^{\mathbb{R}}\) for which \(g+G\subset {\mathcal F}\). The inequality A(HF)\(\leq\omega\) has been proved by \textit{K.~Płotka} [Proc. Amer. Math. Soc. 131, No.~4, 1031--1041 (2003; Zbl 1012.15001)]. The proof of the opposite inequality is based on the fact that there exists a Hamel function which is 3-continuous (i.e., can be covered by three partial continuous functions). This result has been proved by \textit{K.~Płotka} and \textit{I.~Recław} in [``Finitely continuous Hamel basis'', to appear in Real Anal. Exch. 30, No.~2 (2004/2005)].