Chromatic number of finite and infinite graphs and hypergraphs

First we state a few simple finite problems . Chromatic graphs attracted the attention of mathematicians first of all because of the four colour problem, but it was soon realized that interesting and difficult questions can be proved on chromatic graphs which have nothing to do with the four colour problem (or Appel and Haken theorem) . Tutte (and independently Ungar, Zykov and Mycielski) proved that for every k there are k-chromatic graphs without a triangle . First of all, we study sparse graphs of high chromatic number. Erdős proved that for every k and l there is a k -chromatic graph of girth >1 . More precisely, there is a graph on n vertices of girth > l, the largest independent set of which is < n'-"' for a certain absolute constant c (independent of l and n) . Thus the chromatic number is > n`' . For related problems and results see [1-3] . Now we state a few extremal problems . Denote by c(3, n) the smallest integer c for which there is a graph on c vertices of chromatic number n which has no triangle . c(3, 3) = 5 is trivial and it is known that c(3, 4) = 11 ; as far as we know c(3, 5) has not yet been determined or at least is not known to us . It would not be hard to determine c(3, 5) but the determination of c(3, n) for larger values of n will be very difficult . It is not even trivial to prove that c(3, n)/n -> . This was first shown by Erdős who in fact proved c(3, n)> n"" and also r(3, n)> n", r(3, n) is the smallest integer for which every graph of r(3, n) vertices either has a triangle or an independent set of n vertices. The sharpest results known at present are