Does there exist a big graph with that property?

Question

Does there exist a graph with the chromatic number greater than $2013$ and all the cycles of length greater than $2013 $ too?

Answer 1

Yes. This can be proved by probabilistic method, which was first introduced by Paul Erdos. See here for example.

Answer 2

Yes, for every $k$ and every $l$ there are graphs with chromatic number $>k$ And girth $>l$. This is a well-known theorem, usually proved using "the probabilistic method".

See for example this link