On Extremal Graphs with Bounded Girth

By the extremal number ex ( n ; t ) = ex ( n ; { C 3 , C 4 , … , C t } ) we denote the maximum size (number of edges) in a graph of n vertices, n > t , and girth (length of shortest cycle) at least g ⩾ t + 1 . In 1975, Erdős proposed the problem of determining the extremal numbers ex ( n ; 4 ) of a graph of n vertices and girth at least 5. In this paper, we consider a generalized version of this problem, for t ⩾ 5 . In particular, we prove that ex ( n ; 6 ) for n = 29 , 30 and 31 is equal to 45, 47 and 49, respectively.