### On the size of induced acyclic subgraphs in random digraphs

*Joel Spencer, C. R. Subramanian*

#### Abstract

Let D ∈

**D**(n,p) denote a simple random digraph obtained by choosing each of the (n choose 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that
mas(D) ≤ 2 (ln np + 3e)/(ln (1-p)^{-1})

mas(D) = (2 ln np)/ln (1-p)^{-1})
( 1 ±o(1) ).

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