We study the asymptotic behavior of the maximum number of directed cycles of a given
length in a tournament: let c(l) be the limit of the ratio of the maximum number of
cycles of length l in an n-vertex tournament and the expected number of cycles of
length l in the random n-vertex tournament, when n tends to infinity. It is well-known
that c(3) = 1 and c(4) = 4/3. We show that c(l) = 1 if and only if l is not divisible
by four, which settles a conjecture of Bartley and Day. If l is divisible by four,
we show that 1 + 2 center dot (2/pi)(l) <= c(l) <= 1 + (2/pi+ o(1))(l) and determine
the value c(l) exactly for l = 8. We also give a full description of the asymptotic
structure of tournaments with the maximum number of cycles of length l when l is not
divisible by four or l is an element of{4, 8}. (c) 2022 Elsevier Inc. All rights reserved.
