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

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into c_0(\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 $L_p(\mu)$, $1\le p\le2$) 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 $X=L_p(\mu)$, $1\le p<\infty$, bi-Lipschitz embeds into c_0(\Ga) if and only if \dens X<\om_\om