On digraphs without antidirected cycles

Let t ( n ) denote the greatest number of arcs in a diagraph of orders n which does not contain any antidrected cycles. We show that [16/5( n − 1)] ≤ t ( n ) ≤ 1/2 ( n − 1) for n ≥ 5. Let t r ( n ) denote the corresponding quantity for r ‐colorable digraphs. We show that [16/5( n − 1)] ≤ t 5 ( n ) ≤ t 6 ( n ) ≤ 10/3( n − 1) for n ≥ 5 and that t 4 ( n ) = 3( n − 1) for n ≥ 3.