Ramsey numbers involving graphs with large degrees

Let F be a graph of order v ( F ) ≥ 3 and size e ( F ) , and let ρ ( F ) = ( e ( F ) − 1 ) / ( v ( F ) − 2 ) . It is shown that if G n is a graph of order n with average degree d n ≥ 2 , then r ( F , G n ) ≥ c ( d n log d n ) ρ ( F ) for all n , where c = c ( F ) > 0 is a constant.