UEFA 챔피언스리그 대진의 불공정 클러스터, 케일리 그래프가 해결한다

A tight little cluster hidden in the 2025-26 Champions League draw shows why the “new format” needs one small tweak, and gives a beautiful excuse to talk about symmetric graphs. TL;DR: UEFA’s new league-phase format can accidentally produce small clusters of teams that play each other disproportionately often, hurting their collective chance of advancing all at once. A single symmetric graph on 36 vertices, C(36; ±{1, 4, 10, 17}) , eliminates this risk provably. Deploying it needs one small tweak to the country rule for UCL/UEL, and no tweak at all for UECL. Napoli, Benfica, Chelsea, Ajax, Qarabağ . In the 2025-26 UEFA Champions League, they are scheduled to play each other in nine of the ten possible pairings . Only Napoli-Ajax is missing from a complete round-robin among the five. The other nine (Napoli-Benfica, Benfica-Chelsea, Chelsea-Ajax, Ajax-Qarabağ, and five more) all happen in the league phase. UCL 2025-26 - 9 of 10 possible games scheduled. That’s an oddly tight little cluster in a 36-team competition. And it’s the kind of thing that isn’t supposed to happen anymore. Except it does, every season, in almost every UEFA continental competition. In UCL 2024-25 (the first year of the new format), Bayer Leverkusen, Atlético Madrid, Feyenoord, RB Salzburg and Sparta Prague play each other 9 of 10 times. In UEL 2024-25, Roma, Tottenham, AZ, Galatasaray, and IF Elfsborg do the same. UEL 2025-26 has a maximum 5-team cluster of 8 out of 10; UECL 2025-26 has 7 out of 10. UCL 2024-25 — 9 of 10 UEL 2024-25 — 9 of 10 UEL 2025-26 — 8 of 10 UECL 2025-26 — 7 of 10 Click to expand: full stats across all five competitions **How many dense clusters, per season?** Season 5-team clusters with ≥7 matches ≥8 matches =9 matches 4-team clusters with ≥5 matches =6 matches UCL 2024-25 225 17 3 84 4 UCL 2025-26 332 29 1 109 5 UEL 2024-25 240 17 2 87 3 UEL 2025-26 146 6 0 60 0 UECL 2025-26 (6 matches, not 8) 35 0 0 14 0 Two things jump out: (i) every completed UCL/UEL season has produced a cluster with 8 or more internal matches, and (ii) 2025-26 UCL is actually the season with the most 8+ clusters we have on record (29 subsets), though only one hits 9. UECL is meaningfully less dense because it uses a sparser structure (6 opponents per team, not 8). What UEFA changed in 2024 For decades, the Champions League and its sister competitions used the same shape: 32 teams, 8 groups of 4, a round-robin within each group. Each team played 6 group-stage matches (three opponents faced twice each) before the top two per group advanced. In 2024-25, UEFA scrapped that. In its place: a single 36-team “league phase” where each team plays 8 different opponents (2 from each of 4 seeded pots), and a single 36-team table sorts everyone into three buckets: top 8 auto-advance to the round of 16, teams ranked 9-24 go to a knockout playoff, teams 25-36 are eliminated. The same format runs across all three UEFA men’s competitions: Champions League (UCL), Europa League (UEL), Conference League (UECL). The sales pitch was reasonable. More marquee matches. More competitive standings. No more “dead rubber” final matchday when both advancing teams have qualified already. And the “single table” framing removes the arbitrary group draw, which had famously produced imbalanced “groups of death” for years. But it introduced a subtler risk that nobody flagged at the time. Hidden clusters Here’s the thing about 36 teams each playing 8 opponents: it produces a huge network of 144 matches, and inside that network you can find small clusters of teams that happen to play each other disproportionately often. Not because anyone designed it that way; because the draw is random and 144 edges among 36 teams have to land somewhere. Think of the league phase as a graph: 36 nodes , one per team; 144 edges , one per scheduled match. Every team-node touches exactly 8 edges (its 8 opponents). A dense cluster is a small clump of nodes with an unusually large number of edges inside the clump. Here’s what the actual 2025-26 UCL looks like as one such graph: UCL 2025-26 as a graph. 36 team-nodes around the ring, grouped by pot; 144 match-edges drawn as chords. The 9 red edges are the internal pairings of the Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster. Want to poke at the graph yourself? Here’s an interactive viewer for all five past draws (UCL, UEL, UECL, 2024–26) with re-layout, pot/country coloring, and cluster highlighting. The Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster is the extreme case. Of the 376,992 different ways to pick 5 teams out of 36, the actual UCL 25-26 draw has 332 different 5-team clusters with 7 or more internal matches, 29 with 8 or more, and exactly one (the Qarabağ cluster) with 9 , the unique maximum. Why does that matter? Here’s the arithmetic. Each of the 5 cluster teams plays 8 matches, giving the cluster a total of 5 × 8 = 40 team-match slots across the season. Nine of those matches are internal (both teams from the cluster), and each internal match uses up 2 slots (one per participant). So internal matches consume 2 × 9 = 18 slots , leaving 22 slots for matches against teams outside the cluster. Each match distributes at most 3 points (all to the winner) or 2 (a draw). If all 40 slots were external, the cluster could collect up to 40 × 3 = 120 points worth of “point-mining” opportunities against outsiders. But an internal match doesn’t work that way: the 3 winner-points have to go to a cluster team, and the other cluster team gets 0 (or in a draw, both get 1). Points don’t leave the cluster. In short: each internal match trades what could be 6 collectible points against outsiders for 3, or 2, redistributed points within the cluster. The cluster’s collective point ceiling drops from 120 (with 0 internal matches) to 120 − 3 × 9 = 93 points (with 9 internal matches). Divided among 5 teams, that’s an average of 18.6 points per team. In principle enough; top-24 cutoff hovers around 8-11 points. But that’s just the ceiling. Reality is worse. If you’d rather skip the math and simulations, jump directly to what changes under the fix ↓ . Someone in the cluster always pays Beyond the point-ceiling arithmetic above, internal matches also correlate the cluster teams’ fortunes negatively. Every match has a winner and a loser (or two draws), so if Napoli does well internally then someone else in the cluster is doing badly. For “all 5 teams jointly reach the playoff cutoff” to happen, no team can be below the cutoff, and internal matches actively work against that. The cleanest way to see this in a number is a controlled experiment: take five equally-strong teams (just assume for a moment), drop them into the actual 2025-26 UCL field, and vary only the number of matches they play against each other from 0 up to the maximum of 10 (a 5-clique; every one of the 10 possible pairings among the five is scheduled). Their joint chance of all reaching top-24 responds like this (20,000 Monte Carlo sims per cell): Read that as a per-internal-match tax on the cluster: each additional internal pairing costs the group roughly 0.6-0.7 percentage points of joint fate. Going from a fully external schedule to a fully internal one takes joint P from about 17% down to 10%. That’s the pure structural effect of density; team strengths are held equal. (m = 10 hasn’t happened in any real UCL/UEL draw yet; the max observed is 9. But nothing in UEFA’s rules forbids it, so it belongs in the range we’re testing.) Real clusters are worse, because their members aren’t equally strong. Napoli, Benfica, Chelsea, Ajax, and Qarabağ have very different strengths (e.g., clubelo.com ratings); Qarabağ in particular sits well below the other four, and “all 5 advance” is bottlenecked by the weakest member. Re-running the simulation with each team’s actual Elo, the joint probability that all five reach top-24 drops to about 1.4% , versus 4.2% for a strength-matched 5-team group drawn at a typical position in the actual draw (roughly 2 internal mat