Characterizing Large Clique Number in Tournaments

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.

💡 Research Summary

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The paper investigates the “clique number” of a tournament, a concept introduced by Aboulker, Aubian, Charbit, and Lopes (2023). For a tournament T, the back‑edge graph B(T, <) is obtained by fixing a linear ordering < of the vertices and turning every arc that goes against the order into an undirected edge. The tournament’s clique number → ω(T) is defined as the minimum clique number among all possible back‑edge graphs of T. This parameter behaves similarly to the ordinary graph clique number and satisfies → ω(T) ≤ → χ(T), where → χ(T) is the dichromatic number of T.

Two infinite families of tournaments are central to the work:

• Aₙ – the “mountain” family defined recursively by Kim and Kim (2018). Starting from a single vertex A₁, each Aₙ contains n distinguished vertices v₁,…,vₙ together with n − 1 copies of Aₙ₋₁. The orientation is such that vⱼ ⇒ vᵢ for i < j, each copy of Aₙ₋₁ dominates later copies, and the v‑vertices dominate appropriate copies. The back‑edge graph of Aₙ looks like a layered mountain (Figure 1).

• Dₙ – the “Δ‑family” introduced by Berger, Choromanski, Chudnovsky, Fox, Loebl, Scott, Seymour, and Thomassé (2022). D₁ is a single vertex and Dₙ is obtained by taking three subtournaments Dₙ₋₁, Dₙ₋₁, D₁ and arranging them cyclically so that each part is out‑complete to the next (notation Δ(T₁,T₂,T₃)). Dₙ has 2ⁿ⁻¹ vertices.

Lemma 2 shows that both families have unbounded → ω: → ω(Dₙ) grows at least logarithmically in n, while → ω(Aₙ) tends to infinity because → χ(Aₙ)=n and → χ(T) ≤ 9·→ ω(T) (Aboulker et al.).

The main theorem (Theorem 3) states that for every integer n there exists a constant cₙ such that any tournament that avoids both Aₙ and Dₙ has → ω bounded by cₙ. Consequently, a set H of forbidden tournaments forces bounded clique number if and only if it contains some Aₙ and some Dₙ (Corollary 4). This mirrors the known dichromatic‑number characterization for H‑free tournaments (Berger et al.) and narrows the open problem of characterizing H‑free tournaments with bounded dichromatic number to the case of bounded → ω.

The proof proceeds in three conceptual stages.

1. Mountains. An r‑mountain is a recursively defined substructure whose size is at most (r!)² (Lemma 9). Lemma 10 shows that any 2‑coloring of an r‑mountain yields either a red a‑mountain or a blue b‑mountain with a + b = r + 1. Lemma 11 then proves that the presence of an r‑mountain forces → ω(T) ≥ ⌊log₂ r⌋. This establishes that a tournament with large → ω must contain a vertex whose out‑neighbourhood also has large → ω.

2. Bag‑chains. The authors attempt to grow a copy of Dₙ inside T. If the process stalls, a “bag‑chain” emerges: a large subtournament in which most arcs are oriented consistently. By iteratively extending Dₙ and analyzing the failure points, they show that either a full Dₙ appears or a maximal bag‑chain is obtained, and the rest of the tournament attaches to this chain in a highly controlled way.

3. Bounding via Aₙ. In the maximal bag‑chain scenario, the remaining vertices are partitioned into four bag‑chains. The authors then invoke the Aₙ‑family to bound the clique number of each bag‑chain, because Aₙ is precisely the structure that can certify large → ω within a relatively small vertex set. Combining these bounds yields Theorem 8: there exists a function f such that → ω(T) < f(ω_A(T)+ω_D(T)), where ω_A(T) and ω_D(T) denote the largest n for which T contains Aₙ or Dₙ respectively.

Corollary 6 parallels a result of Harutyunyan, Le, Thomassé, and Wu (2020) for dichromatic number, showing that → ω(T) is bounded by a function of the maximum → ω of the out‑neighbourhoods of individual vertices. Finally, the paper resolves Conjecture 5.8 of Aboulker et al., proving that large → ω is always certified by a bounded‑size subtournament (Corollary 7). This contrasts sharply with dichromatic number, where such a bounded‑size certification fails.

Overall, the work provides a clean structural dichotomy for tournaments with large clique number: any such tournament must contain a large Aₙ or a large Dₙ, and consequently the clique number can always be witnessed by a small, well‑understood subconfiguration. This advances the theory of tournament invariants and opens new avenues for studying unavoidable subtournaments in the context of both clique and dichromatic numbers.