Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\text{Ga})$

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(\text{Ga})$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(\mu )$, $1\le p\le 2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(\mu )$, $1\le p<\mathrm{\infty }$, bi-Lipschitz embeds into $c_0(\text{Ga})$ if and only if $\text{dens}X<{\text{om}}_{\text{om}}$.