The Shifted Turán Sieve Method on Tournaments

Abstract

We construct a shifted version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted $r$-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.

The shifted Turán sieve method on tournaments II

“…For this, we need to first discuss how cycles b 1 and b 2 intersect each other, and this intersection can have a very complicated structure. In this paper, we use a counting method developed in [5] for estimating the sum of n(b 1 , b 2 ), that consists of "omit some existing cases" and "include some non-existing cases" to get the expected main contribution. Then we compare the "under-counting" and "over-counting" of the main contribution to get the correct estimate.…”

“…This method was further generalized to a bipartite graph in [7] to investigate several combinatorial questions. In a previous paper [5], we constructed a shifted version and used it to bound the number of tournaments with few r-cycles.…”

“…Let G = (A, B, E) be a bipartite graph with finite partite sets A, B and edge set E. The Turán sieve [7, Corollary 1] and its shifted version [5,Corollary 1.3] can be easily deduced from the theorem above.…”

“…In [5], the authors applied the shifted Turán sieve method to establish upper bounds for the number of tournaments and t-partite tournaments with a given number of vertices and exactly k restricted r-cycles, where k is fixed and 3 ≤ r ≤ t ≤ n. Joining the result obtained there with the sieve improvement, it is possible to obtain similar bounds, but allowing more cycles than a fixed quantity. The case of unrestricted cycles on t-partite tournaments is much harder than the restricted case, because the cycles can return to the same partition but cannot selfintersect, and this can cause a break in symmetry.…”

“…The bound presented in Theorem 1.4 is general but not tight, in the sense that it holds for any cycle length, however the upper bound can be greatly improved. Obtaining the exact bound for the problems presented here and in [5] is an interesting and also a very difficult problem. Yet, using other combinatorial arguments, it was possible to give the precise bound for the number of tournaments with a "small" number of 3-cycles (Theorem 1.5).…”