Local and global colorability of graphs

It is shown that for any fixed c ? 3 and r , the maximum possible chromatic number of a graph on n vertices in which every subgraph of radius at most r is c -colorable is ? ? ( n 1 r + 1 ) : it is O ( ( n / log n ) 1 r + 1 ) and ? ( n 1 r + 1 / log n ) . The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random n -vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius r in it are 2-degenerate.