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On the maximum cardinality of a consistent set of arcs in a random tournament. (English) Zbl 0531.05036
Let $$f(T_ n)$$ denote the maximum number of arcs possible in an acyclic subgraph of a random tournament $$T_ n$$. The author shows that $$f(T_ n)<n(n-1)/4+1.73n^{3/2}$$ with probability tending to one as n tends to infinity, thereby sharpening a result of J. Spencer [Period. Math. Hung. 11, 131-144 (1980; Zbl 0349.05011)].
Reviewer: J.W.Moon

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C80 Random graphs (graph-theoretic aspects) 60C05 Combinatorial probability
##### Keywords:
acyclic subgraph; random tournament
Full Text:
##### References:
 [1] Chernoff, H, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. math. statist., 23, 493-509, (1963) [2] Erdös, P; Moon, J.W, On sets of consistent arcs in a tournament, Canad. math. bull., 8, 269-271, (1965) · Zbl 0137.43301 [3] Erdös, P; Spencer, J, () [4] Spencer, J, Optimal ranking of tournaments, Networks, 1, 135-138, (1971) · Zbl 0236.05110 [5] Spencer, J, Optimally ranking unrankable tournaments, Period. math. hungar., 11, 2, 131-144, (1980) · Zbl 0349.05011
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