A new arXiv preprint reports a proof of a universal lower bound for discrepancy between pairs of finite tournaments. For every pair of order n, meaning each tournament has n vertices, the overall discrepancy is at least an absolute constant times n to the three-halves power. The result answers the paper's central question affirmatively for every order n of at least 2, rather than only for a selected family of examples.
Here, discrepancy is a mathematical measure with two one-sided forms: positive and negative. Overall discrepancy is the larger of the two. The paper also proves that the product of those one-sided quantities is at least an absolute constant times n cubed. That product bound sits behind the headline conclusion about the overall discrepancy.
A proof built around relabelling
To build the argument, the proof represents tournaments with skew-symmetric tournament matrices. It studies relabellings of those matrices, including uniformly random permutations of the labels. Averaged over a uniform random permutation, the normalized matrix correlation is exactly zero. The proof also includes an auxiliary estimate for sums formed from random permutations.
The key bridge is a transposition-amplification lemma. It links a local fluctuation parameter, written as gamma, to the product of the positive and negative discrepancies. The lemma says the product of the positive and negative discrepancies is at least a constant times n squared times gamma squared, and it states that its constant is 10 to the minus fifth power. This is the step that carries a local fluctuation into the global discrepancy product.
The proof then establishes the size of gamma. For every pair of tournament matrices of order n at least 4, the local fluctuation is bounded below by an absolute constant times the square root of n. That constant is asserted to be absolute but is not given numerically. The argument also shows that at least one-thirteenth of ordered vertex pairs are good in every tournament of order n at least 4. This proportion is a theorem over all ordered pairs, not an empirical estimate.
Why the exponent matters
The paper calls the exponent 3/2 optimal, but that conclusion rests on an existence benchmark. Some tournaments, when paired with the transitive tournament on n vertices, have discrepancy bounded above at order n to the three-halves power. These examples show why a universal lower bound cannot demand a larger power of n in the same general form. They do not provide a universal upper bound of that order for every tournament pair.
One important qualification is numerical. The final theorem asserts an absolute constant, but the supplied analysis does not give one fully numerical value for it. The same issue applies to the auxiliary constants used in the proof, which are not all specified numerically.
Preprint status and disclosures
The work is an arXiv version 1 preprint dated 28 August 2026. The authors report support from the National Key R&D Program of China and the National Natural Science Foundation of China.
The acknowledgements also disclose using ChatGPT 5.6 Pro to discuss proof strategies, check proofs and improve exposition.
Paper data and sources
Original title: Intersections of Tournaments
Authors: Zhanping Yang, Qinghou Zeng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text