A tight small cluster hidden successful the 2025-26 Champions League tie shows why the “new format” needs 1 mini tweak, and gives a beautiful excuse to talk astir symmetric graphs.
TL;DR: UEFA’s caller league-phase format tin accidentally nutrient mini clusters of teams that play each different disproportionately often, hurting their corporate chance of advancing each astatine once. A azygous symmetric chart connected 36 vertices, C(36; ±{1, 4, 10, 17}), eliminates this consequence provably. Deploying it needs 1 mini tweak to the state norm for UCL/UEL, and nary tweak astatine each for UECL.
Napoli, Benfica, Chelsea, Ajax, Qarabağ.
In the 2025-26 UEFA Champions League, they are scheduled to play each different successful nine of the 10 imaginable pairings. Only Napoli-Ajax is missing from a complete round-robin among the five. The different 9 (Napoli-Benfica, Benfica-Chelsea, Chelsea-Ajax, Ajax-Qarabağ, and 5 more) each hap successful the convention phase.
UCL 2025-26 - 9 of 10 imaginable games scheduled.
That’s an oddly tight small cluster successful a 36-team competition. And it’s the benignant of point that isn’t expected to hap anymore.
Except it does, each season, successful almost each UEFA continental competition. In UCL 2024-25 (the first twelvemonth of the caller format), Bayer Leverkusen, Atlético Madrid, Feyenoord, RB Salzburg and Sparta Prague play each different 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 retired of 10; UECL 2025-26 has 7 retired 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 |
| 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 |
What UEFA changed successful 2024
For decades, the Champions League and its sister competitions utilized the aforesaid shape: 32 teams, 8 groups of 4, a round-robin wrong each group. Each squad played 6 group-stage matches (three opponents faced doubly each) earlier the apical 2 per group advanced.
In 2024-25, UEFA scrapped that. In its place: a azygous 36-team “league phase” wherever each squad plays 8 different opponents (2 from each of 4 seeded pots), and a azygous 36-team array sorts everyone into 3 buckets: apical 8 auto-advance to the information of 16, teams classed 9-24 spell to a knockout playoff, teams 25-36 are eliminated. The aforesaid format runs crossed each 3 UEFA men’s competitions: Champions League (UCL), Europa League (UEL), Conference League (UECL).
The income transportation was reasonable. More marquee matches. More competitory standings. No much “dead rubber” last matchday erstwhile some advancing teams person 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 consequence that cipher flagged astatine the time.
Here’s the point astir 36 teams each playing 8 opponents: it produces a immense web of 144 matches, and wrong that web you tin find mini clusters of teams that hap to play each different disproportionately often. Not because anyone designed it that way; because the tie is random and 144 edges among 36 teams person to onshore somewhere.
Think of the convention shape arsenic a graph: 36 nodes, 1 per team; 144 edges, 1 per scheduled match. Every team-node touches precisely 8 edges (its 8 opponents). A dense cluster is simply a mini clump of nodes pinch an unusually ample number of edges inside the clump.
Here’s what the existent 2025-26 UCL looks for illustration arsenic 1 specified graph:
UCL 2025-26 arsenic a graph. 36 team-nodes astir the ring, grouped by pot; 144 match-edges drawn arsenic chords. The 9 reddish edges are the soul pairings of the Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster.
Want to flick astatine the chart yourself? Here’s an interactive spectator for each 5 past draws (UCL, UEL, UECL, 2024–26) pinch re-layout, pot/country coloring, and cluster highlighting.
The Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster is the utmost case. Of the 376,992 different ways to prime 5 teams retired of 36, the existent UCL 25-26 tie has 332 different 5-team clusters pinch 7 aliases much soul matches, 29 pinch 8 aliases more, and precisely 1 (the Qarabağ cluster) pinch 9, the unsocial maximum.
Why does that matter?
Here’s the arithmetic. Each of the 5 cluster teams plays 8 matches, giving the cluster a full of 5 × 8 = 40 team-match slots crossed the season. Nine of those matches are soul (both teams from the cluster), and each soul lucifer uses up 2 slots (one per participant). So soul matches devour 2 × 9 = 18 slots, leaving 22 slots for matches against teams extracurricular the cluster.
Each lucifer distributes astatine astir 3 points (all to the winner) aliases 2 (a draw). If each 40 slots were external, the cluster could cod up to 40 × 3 = 120 points worthy of “point-mining” opportunities against outsiders. But an soul lucifer doesn’t activity that way: the 3 winner-points have to spell to a cluster team, and the different cluster squad gets 0 (or successful a draw, some get 1). Points don’t time off the cluster.
In short: each soul lucifer trades what could beryllium 6 collectible points against outsiders for 3, aliases 2, redistributed points wrong the cluster. The cluster’s corporate constituent ceiling drops from 120 (with 0 soul matches) to 120 − 3 × 9 = 93 points (with 9 soul matches).
Divided among 5 teams, that’s an mean of 18.6 points per team. In rule enough; top-24 cutoff hovers astir 8-11 points. But that’s conscionable the ceiling. Reality is worse.
If you’d alternatively skip the mathematics and simulations, jump straight to what changes nether the hole ↓.
Someone successful the cluster ever pays
Beyond the point-ceiling arithmetic above, soul matches besides correlate the cluster teams’ fortunes negatively. Every lucifer has a victor and a loser (or 2 draws), truthful if Napoli does good internally past personification other successful the cluster is doing badly. For “all 5 teams jointly scope the playoff cutoff” to happen, no squad tin beryllium beneath the cutoff, and soul matches actively activity against that.
The cleanest measurement to spot this successful a number is simply a controlled experiment: return 5 equally-strong teams (just presume for a moment), driblet them into the existent 2025-26 UCL field, and alteration only the number of matches they play against each different from 0 up to the maximum of 10 (a 5-clique; each 1 of the 10 imaginable pairings among the 5 is scheduled). Their associated chance of all reaching top-24 responds for illustration this (20,000 Monte Carlo sims per cell):
Read that arsenic a per-internal-match taxation connected the cluster: each further soul pairing costs the group astir 0.6-0.7 percent points of associated fate. Going from a afloat outer schedule to a afloat soul 1 takes associated P from astir 17% down to 10%. That’s the axenic structural effect of density; squad strengths are held equal.
(m = 10 hasn’t happened successful immoderate existent UCL/UEL tie yet; the max observed is 9. But thing successful UEFA’s rules forbids it, truthful it belongs successful the scope we’re testing.)
Real clusters are worse, because their members aren’t arsenic strong. Napoli, Benfica, Chelsea, Ajax, and Qarabağ person very different strengths (e.g., clubelo.com ratings); Qarabağ successful peculiar sits good beneath the different four, and “all 5 advance” is bottlenecked by the weakest member. Re-running the simulation pinch each team’s actual Elo, the associated probability that each 5 scope top-24 drops to astir 1.4%, versus 4.2% for a strength-matched 5-team group drawn astatine a emblematic position successful the existent tie (roughly 2 soul matches connected average, versus the Qarabağ cluster’s 9). That’s a 68% comparative reduction successful associated fate, purely from the structural density of their pairings.
One important asymmetry: dense clusters tin only shape among teams that are really allowed to play each other. UEFA’s state norm bans same-country matchups, so, say, 2 English teams can never onshore successful the aforesaid cluster pinch each other. Countries pinch galore teams are automatically shielded from ever being co-clustered internally. The consequence of getting sucked into a Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster truthful falls disproportionately connected teams from mini federations.
The constituent isn’t that this circumstantial tie was cursed; it’s somewhat supra mean successful cluster density (roughly apical 20% of comparable random draws we simulated), not statistically extreme. It’s that the format allows this. Any of the 3 competitions tin nutrient a Napoli-Benfica-Chelsea-Ajax-Qarabağ for illustration business successful immoderate early season. It’s a structural consequence of moving a random tie complete a chart this dense. So the mobility becomes: is location a measurement to destruct that consequence by design, alternatively than hoping the random tie stays kind?
The fairness principle: cipher sits successful a worse spot than anyone else
Here’s the fairness question, framed a spot abstractly. In the existent UCL 25-26 draw, Qarabağ ended up sitting successful the mediate of the densest 5-team cluster. Pafos didn’t. Neither squad chose their opponents; that was decided by the random draw. But the positions they landed successful were structurally different: Qarabağ’s force group overlaps a batch pinch different teams’ force sets (they stock opponents), while Pafos’s overlaps less.
To beryllium clear, “Qarabağ sits successful a dense cluster” is not the aforesaid arsenic “Qarabağ will decorativeness last”. Qarabağ (Pot 4) had a genuinely good 2024-25 run nether the caller format; azygous teams hit their expected decorativeness each the time. What the cluster study says is that the five of them jointly are little apt to all advance; immoderate subset almost surely will.
In fact, look backmost astatine past year’s analogous 5-team cluster successful UCL 24-25 (Leverkusen, Atlético, Feyenoord, Salzburg, Sparta): 3 of them made top-24 (Leverkusen, Atlético, Feyenoord); the different 2 (Salzburg and Sparta) didn’t. And of the existent Pot-3 brace Ajax + Napoli, some missed top-24 past play successful akin tight-cluster situations. The shape is: dense cluster ⇒ immoderate cluster squad pays the price; which peculiar squad is simply a matter of individual form. The unfairness is at the cluster level, distributed crossed the affected teams.
You can’t blasted the tie for “picking bad teams to play”; each squad is arsenic apt to tie immoderate opponent; but you tin ask: what if the graph of positions were designed truthful that nary position is worse than immoderate other?
In chart theory, this is called vertex-transitivity. A chart is vertex-transitive if you tin descent immoderate vertex onto immoderate different vertex and the image looks unchanged. Every vertex has the aforesaid number of neighbors (that’s conscionable being “regular”), but besides the aforesaid number of 2-hop neighbors, the aforesaid number of 3-cycles done it, the aforesaid number of 4-cycles, the aforesaid everything. If your squad is astatine “position 7” alternatively of “position 12,” your structural business is identical; only the circumstantial identities of your opponents change.
That’s a beardown fairness property. Draws would still beryllium random (which circumstantial squad lands astatine which vertex depends connected the balls that travel out), but nary squad could kick that their position was worse than different team’s. Only their opponents’ strengths; which is the benignant of luck fans accept.
The math: Cayley graphs
How do you conception a vertex-transitive chart connected 36 vertices wherever each squad has 8 opponents divided crossed 4 pots the measurement UEFA wants?
One classical answer: Cayley graphs. Take the integers modulo 36 arsenic your vertex set; deliberation of them arranged astir a timepiece look pinch 36 tick marks. Pick a mini group of “offsets”.
For a first illustration, effort the offset group {±1, ±4, ±6, ±11}. Connect each vertex one to the 8 vertices astatine one ± 1, one ± 4, one ± 6, one ± 11 (mod 36).
Because the aforesaid offsets use to each vertex, each vertex has the aforesaid section vicinity by construction. Shift-invariant. Perfectly vertex-transitive.
Now statement that up pinch UEFA’s cookware structure. Partition the 36 vertices by their residue mod 4: vertices 0, 4, 8, …, 32 are “pot A”; vertices 1, 5, 9, …, 33 are “pot B”; and truthful on. There are 9 vertices successful each pot, matching UEFA’s 4 pots of 9.
For each offset, we tin fig retired which cookware it lands you in. Offset ±4 keeps you successful the aforesaid cookware (within-pot opponent). Offset ±6 puts you successful an adjacent cookware (residue 2 mod 4 gives a “shift by 2 pots”). Offset ±1 aliases ±11 puts you successful an adjacent pot. Pick offsets cautiously and each vertex ends up pinch 2 opponents successful each pot: precisely what UEFA requires.
So a properly-chosen offset group gives us: 8 opponents per team, 2 from each pot, perfectly vertex-transitive, algorithmically simple. That’s the full idea. But not each offset group is good; immoderate create dense clusters moreover worse than the existent UCL draw. Now we conscionable person to find the best group of offsets.
Which offsets are best?
Vertex-transitivity guarantees adjacent positions, but not automatically a sparse graph, truthful we want to prime the offset group that minimizes section density. The first barroom is triangle elimination: nary 3 teams among the 36 each play each different (a triangle is the smallest imaginable dense cluster). Triangles tin beryllium eliminated wholly for a subset of offset choices. Four-cycle elimination would beryllium beautiful too, but it’s provably intolerable connected 36 vertices (Moore’s bound says an 8-regular chart pinch nary 4-cycles needs astatine slightest 1 + 8 + 8·7 = 65 vertices); the champion we tin do is minimize really galore 4-cycles each squad sits inside.
There are 432 valid choices of offset sets that fulfill the UEFA cookware structure. Of those, only 21 chopped graphs stay aft we propulsion retired thing pinch triangles and illness mathematically-equivalent candidates. We classed those 21 connected 5 fairness metrics, and each 5 constituent astatine the aforesaid winner: C(36; ±{1, 4, 10, 17}).
Click to expand: the fairness metrics we tested and the winner's numbers The 5 metrics, successful plain terms: 1. **Fewest short cycles per team**: really tangled is your section neighborhood, astatine rhythm lengths 4, 5, 6, and up. 2. **Most azygous "shared opponents" distribution**: nary team's opponents are unusually intertwined pinch immoderate different team's. 3. **Fewest dense 5-team clusters**: count of 5-team subsets that scope the maximum internal-match count allowed by the chart structure. 4. Uniformly-weighted sum of rhythm counts. 5. Small-cycle-heavier weighted sum of rhythm counts. All 5 agree. What the victor buys you: - Only 42 four-cycles per squad (compared to 49-132 for different candidates). - Standard deviation of shared-opponents is 0.94 (next lowest is 1.18; the pathological worst lawsuit has 2.75). - No five-team cluster successful the chart exceeds 6 soul matches. Ever. And only 252 clusters moreover scope 6 (compared to 360-1080 for the different candidates). - The tightest imaginable "both sides play each of the different side" subgraphs are small: nary specified building has 3+ teams connected some sides.
Here’s the victor graph:
The symmetric chart C(36; ±{1, 4, 10, 17}). Vertex i is connected to i ± 1, i ± 4, i ± 10, i ± 17 (mod 36). Colors people the 4 pots (residue mod 4).
Each vertex has 8 neighbors (i ± 1, one ± 4, one ± 10, one ± 17, each mod 36). The image looks the aforesaid if you rotate it: that’s the vertex-transitivity.
Can it really beryllium deployed?
Almost. There’s 1 wrinkle: for the 2025-26 UCL squad group (6 English teams, 5 Spanish, 4 Italian, 4 German, and 17 others from smaller federations), we tried to delegate the 36 teams to the 36 vertices of C(36; ±{1, 4, 10, 17}) while respecting UEFA’s state norm (“no 2 teams from the aforesaid state play each other, and each squad faces astatine astir 2 opponents from immoderate different country”).
Using Google’s OR-Tools CP-SAT solver, we proved: strictly impossible. There is nary assignment. The state distribution is excessively concentrated successful the apical 4 federations for a vertex-transitive chart to accommodate.
And this isn’t a 2025-26 quirk. The upcoming UCL 2026-27 tie connected Thursday looks fundamentally the same: astir 5 English, 5 Spanish, 4 Italian, 4 German, and perchance 4 French teams. The attraction successful apical federations is simply a imperishable characteristic of modern qualification, not a one-off. So the strict norm is going to support hitting the aforesaid wall each play connected the symmetric graph.
But 2 mini argumentation tweaks each reconstruct feasibility:
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Option A: raise the “max opponents from aforesaid country” headdress from 2 to 3. A one-integer alteration to UEFA’s regulation. Same-country matchups are still forbidden; the alteration is only that a squad whitethorn look up to 3, alternatively than 2, opponents from immoderate azygous foreign country.
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Option B: divided each ample federation (≥4 teams) into 2 sub-groups (e.g., ENG_A = {Chelsea, Man City, Tottenham}, ENG_B = {Liverpool, Arsenal, Newcastle}), past dainty the sub-groups arsenic if they were abstracted countries. Same-sub-group teams still don’t play, but cross-sub-group same-country matches (Liverpool vs. Chelsea) go allowed, capped astatine 2 per sub-group. A knock-on effect: a non-English squad tin now look up to 4 English opponents (2 from each sub-group).
Option A preserves the intent of the state norm (limiting same-federation opponents to a mini constant), whereas Option B tin beryllium considered fairer arsenic the advantage of countries pinch galore teams is reduced. Both make the perfectly symmetric chart deployable.
A unrecorded tie ceremonial nether either action would look identical to, aliases moreover amended than, today’s. The 36 squad balls spell into a drum. Each ball, erstwhile drawn, gets placed astatine 1 of the valid vertex positions utilizing the aforesaid constraint-propagation package UEFA already uses. Between 100 cardinal and 10 cardinal valid assignments exist: plentifulness of randomness for the tie to consciousness unpredictable. Computation is accelerated capable that it’s invisible to viewers: astir 55 milliseconds of solver activity per ball, astir 2 seconds for the full 36-ball draw. The ceremony’s pacing (the balls, the walk-outs, the presenters explaining what conscionable happened) is group by theatrics, not by the compute, and would beryllium unchanged.
And for UECL, they don’t moreover request the norm tweak. The Conference League’s 6-pot / 6-opponent building has capable slack that a symmetric chart absorbs the state norm as-is. The victor successful that mounting is simply a Cayley chart connected the group Z/18 × Z/2 pinch offset group {(1,0), (2,1), (9,0), (9,1), (16,1), (17,0)}; each squad ends up pinch 1 force per pot, nary same-country matchups, astatine astir 2 opponents from immoderate azygous overseas country. Zero regularisation changes required.
Scheduling aft the draw
Assigning teams to positions is 1 half of deployment. The different half is scheduling the resulting matches into matchdays; and present too, the symmetric chart does thing nice.
Once the pairings are known, UEFA still has to slot the 144 matches into 8 matchdays truthful that nary squad plays doubly connected the aforesaid nighttime (!). That’s the scheduling step. There’s a subtle drawback present that Guyon et al. (2025) uncovered: immoderate random tie outcomes (very rare, but mathematically possible) nutrient a lucifer chart that cannot fresh into 8 matchdays and would request 9. After they alerted UEFA of this, UEFA folded the 8-matchday-schedulability cheque into the tie package itself, truthful specified outcomes are prevented upfront alternatively than fixed aft the fact.
Our fixed-graph attack removes the consequence entirely. We verified pinch a constraint solver that:
- The UCL victor chart splits cleanly into 8 matchdays of 18 matches each (7 is provably excessively few).
- The UECL victor chart splits cleanly into 6 matchdays of 18 matches each (5 is provably excessively few).
- Under the earthy home/away norm (“host if you’re the little number modulo 36”), each squad gets precisely 4 location + 4 away successful UCL (3+3 successful UECL), and each matchday is simply a perfectly balanced 18-home / 18-away split. No optimizer needed; it falls retired of the algebra.
Everything other UEFA cares astir (avoiding consecutive location games, spacing delicate fixtures, TV-window rules) is astir the ordering of the matchdays, not the chart itself, truthful their existing scheduler transfers unchanged.
What this changes
Under the existent UEFA format, successful immoderate fixed season, immoderate mini group of teams whitethorn find themselves successful a cluster for illustration the Napoli-Benfica-Chelsea-Ajax-Qarabağ cluster; done nary responsibility of their own. Their associated chances of advancement return a existent deed from a structural mishap of the draw. And crucially, this hits unevenly astatine the federation level. Individually, immoderate squad tin still beryllium pulled into a dense cluster: Liverpool could plausibly onshore successful 1 pinch 4 non-English teammates, conscionable arsenic Qarabağ did. But nary cluster tin incorporate two aliases much teams from the aforesaid country (the same-country prohibition rules that out), truthful the Premier League arsenic a full cannot suffer the compounded deed that comes from having aggregate of its ain teams jointly stuck. Its six clubs are, successful effect, distributed crossed up to six abstracted imaginable clusters alternatively of concentrated successful one. Azerbaijan’s azygous UCL entrant has nary specified protection; immoderate cluster it lands in, the full nationalist practice is exposed. The existent strategy softly rewards federations pinch galore entrants, moreover erstwhile it does not shield immoderate individual team.
Under the symmetric-graph proposal, that can’t happen. Small dense clusters for illustration the Napoli-Benfica-Chelsea-Ajax-Qarabağ pouch are structurally intolerable successful the caller proposal. No squad occupies a structurally worse position than immoderate different team. The only root of asymmetry is force strength; which is simply a benignant of luck fans understand and accept.
Fairness stops being a statistical spot of the random tie and becomes a mathematical spot of the chart itself: provable per-team, for each team, each season. The mathematics is done and public. All that’s near is simply a decision: alteration 1 statement of the state rule, people 1 fixed graph, and clusters for illustration Napoli-Benfica-Chelsea-Ajax-Qarabağ extremity being possible.
UPDATE — the 2026-27 draws are in, pinch 2 cleanable guardant predictions
The 2026-27 league-phase draws for UCL, UEL, and UECL are completed a fewer days ago, and the hidden-cluster shape shows up successful each three. The astir absorbing cases this twelvemonth aren’t the single-densest tuples — they’re two balanced high-strength six-team clusters successful the UCL that make for unusually cleanable guardant tests of the argument.
The first is PSG, Manchester City, Barcelona, Sporting CP, Aston Villa, Galatasaray — 11 of 15 imaginable pairings scheduled among themselves. Split it by pot: the apical 3 (PSG/City/Barcelona) are near-certain to beforehand individually, but the bottommost 3 (Sporting/Villa/Galatasaray) play each different doubly wrong the cluster (Sporting-Gala and Villa-Gala are some scheduled) and each besides plays 2 of the top-three elites. Galatasaray is the doubly-squeezed one, pinch only 4 outer games. Under the density-penalty mechanism, astatine slightest one, apt two, of {Sporting, Villa, Galatasaray} will miss top-24, and Galatasaray is the highest-risk.
The 2nd is Bayern, Arsenal, Dortmund, Real Betis, Lille, Bodø/Glimt — aforesaid 11-of-15 density, aforesaid anatomy. Real Betis draws each 3 elites (Bayern + Arsenal + Dortmund) positive Lille, leaving only 4 outer games. Lille is the doubly-squeezed adjacent present (playing some Betis and Bodø). At slightest 1 of {Betis, Lille, Bodø/Glimt} is astatine existent consequence of missing top-24, pinch Betis and Lille the starring candidates. UEL 2026-27 and UECL 2026-27 besides produced max-density 6-tuples this year, but some person 1 clearly-weakest squad (Lillestrøm successful the UEL cluster; Kairat successful the UECL one), which makes those cases confounded alternatively than cleanable tests of the mechanism.
Two seasons successful a statement now, crossed each 3 UEFA continental competitions, the aforesaid structural consequence keeps producing the aforesaid benignant of small pit location connected the fixture list. It’s not an overseas year. It’s the format. All 3 caller draws are loaded into the interactive tool.
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