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 tr (n) denote the corresponding quantity for r-colorable digraphs. We show that [16/5(n − 1)] ≤ t5(n) ≤ t6(n) ≤ 10/3(n − 1) for n ≥ 5 and that t4(n) = 3(n − 1) for n ≥ 3.