Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f ( n , r ) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph K n 3 are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that f ( n , 5 ) ≥ 5 n 7 and f ( n , 6 ) ≥ 2 n 3 . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).