This problem was effectively solved at the end of the 18th century with Naval Signalling Flags. There were experiments to see which contrasts were visible and which were not. There are no green flags, for example, as the difference between green and blue was hard to spot, particularly with changing lighting at sunset.

The 26 letters allowed you to spell words, but you were also limited by the flags in your locker and the space on your mast. The flags were supplemented by national flags, and telegraphic codes using rarely-used letters. See, for example how the famous England Expects... signal sent by Nelson was a compromise between the message, the flags they had in the signal locker, and the telegraph codes. The upshot is that each of the 26 letters had to be distinguishable in isolation (unlike letters, which can often be guessed from the surrounding ones).

Flags fulfilled much of the requirements for a sports kit. The flags had to be readable in any orientation, and under different lighting conditions.

Having played with it a bit, I'm pretty sure it is possible. I'll give my answer in visual form first, then talk through it –

We can use 7 colours simply, without needing to mix: Black, White, Green, Yellow, Red, Royal blue, Sky blue

I shied away from using orange alone: a tricky colour which is adjacent to both red and yellow. Magenta/pinks are too close to red, and there's no reason not to add a second colour to pink to help the matter.

Maroon alone could work (by 'work' I mean 'have sufficient contrast with red) – take a look –

– but it's better to add a contrasting colour.

Second, I started tapping the idea of contrasting colours. You can have a yellow-and-purple team because that contrasts strongly against both yellow and purple. (You couldn't have a blue-and-purple team.)

This give four extra teams: green&red, orange&blue, yellow&purple, black&white.

I made orange&blue and yellow&purple two different patterns, as they have limited contrast (orange is like yellow, blue is like purple).

We haven't utilised maroon yet. What contrasts with maroon? Blue. So put West Ham in the league.

Red&green are considered opposite colours, maximum contrast on the colour wheel. But red&yellow is also very highly contrasting. This guide to contrast for web design helped me

Looking through the list of 20 colours, I'd skipped brown, beige, olive, and grey as being too weak alone. So throw them all together! And what does that make? Camo, more or less. Add a camo team; that's very different from green.

This is getting close to 100% optimal. The problem is tractable. A colour normally only has one or two potential clashes, so you just need to make it distinct from those with some alteration.

With $N$ colors and $M$ different articles of clothing to color, you can create $N^M$ unique kit combinations.

If the uniform consists of shirts, pants, and shoes, you need only 3 unique colors to have 27 unique color combinations.

If the uniform consists of just shirts and pants, you need 6 colors to yield 36 unique color combinations. We can certainly get 6 unique enough colors (white, black, red, blue, yellow, green), the only question becomes is a red-blue team distinct enough from a blue-red team. If it's not, we can simply double the number of possible kits by adding stripes to all kits to make new kits.

Add in color-able embellishments like stripes and their directionality, and the number of possible kit combinations grows very quickly. 21 unique combinations is quite doable even with relatively few distinct colors and limited options for what to color - the only way it's really not feasible is if the uniform must be a single color.

Below I use the term livery in most places, as this is the most generic terminology for what you’re trying to talk about. It encompass not just the colors, but also the patterns, insignias, and even things like the exact style of outfit.

First, a frame challenge: Alternate livery doesn’t really hurt branding all that much.

I encourage you to look at how alternative livery is actually used in real life. In a majority of cases, a team’s primary livery is used in almost all cases, they just use different colors on their uniforms during a match, and the primary purpose is so that the two teams can be clearly distinguished while playing. While the alternative livery is often distinguished in many sports by being ‘away colors’, or an ‘away uniform’, that distinction only really exists because if a team needs to change livery for a match, it’s by convention always the team that is not hosting.

Yes, there’s some loss of visual distinctiveness when watching a broadcast of a match, and you may have to check the scoreboard or something equivalent to know for sure who is playing, but that’s about it when it comes to the alternate livery hurting branding.

Additionally, alternative livery helps with merchandising (because it gives you a wider variety of things to sell). This may be significant if the teams derive at least part of their funding from selling merchandise.

Second, an actual solution.

Pick seven distinct colors. Each team gets a unique set of three from that list of seven. Problem solved.

There are 35 possible combinations of 3 colors selected out of 7 possibilities, so this scales up to 35 teams (which helpfully lets you ‘retire’ combinations when a teams are disbanded, allowing for better consistency across such transitions). Each team will have a unique set of colors, so the entire rest of the livery, including things like patterns, can actually be identical (though it shouldn’t be identical for a couple of reasons, such as pattern recognition being important for quick identification even if the colors are distinctive).

In terms of distinctiveness, I would probably go with (using the swatches on the site you referenced) white, black, red, yellow, green, blue, and one of orange, cyan, or magenta. That gets you clear visual distinctiveness of each color, while still being reasonably distinctive even for people with color blindness.

Overall though, this requires the league to assign colors as opposed to the teams just picking their own colors, because cultural bias will tend towards specific colors being picked. You’ll notice for example that red is a very common color for sports teams in Western cultures because of it’s associations with feelings of aggression (in the sense of aggressive competitiveness more than violence, though for some sports the violence aspect fits too).

Color combos

Every real sports league has this problem. What most of them do is to have teams pick a primary, secondary, and tertiary color. Jerseys are usually the primary (home jersey) or secondary/tertiary (away jersey). Often one of the secondary or tertiary colors is white or black and mainly used for shadow or extra contrast.

You could also try patterns, like Tartan cloth, which is a similar idea.

Culture is a bigger obstacle than mathematics

Mathematically, this issue can be solved using the combination formula:

C_(n,r) = n!/((n-r)!*r!)

Most people can consistently distinguish by name and on sight 14 colors [Black, Grey, White, Red, Dark-Red(AKA: Maroon or Burgundy), Yellow, Gold, Orange, Brown, Green, Blue-Green (AKA: Teal, Aqua, Cyan, etc.), Blue, Purple, and Pink]. Even nearly color blind people can tell the difference between black, white, red, green, blue, and gold. This gives you somewhere between 6-14 colors to work with depending on how handicap accessible you want your team kits to be. Now let's assume that kits can have anywhere from 1-3 of these colors.

For a 6 color system you get:

C_x = C_(6,1) + C_(6,2) +C_(6,3) = 6!/(1!(6-1)!) + 6!/(1!(6-1)!) + 6!/(1!(6-1)!)

C_x = 5 + 15 + 20

C_x = 40

And for a 14 color system you get:

C_x = C_(14,1) + C_(14,2) +C_(14,3) = 14!/(1!(14-1)!) + 14!/(1!(14-1)!) + 14!/(1!(14-1)!)

C_x = 14 + 91 + 364

C_x = 469

So, by just using colors in combination of 3 or less, you can support a league of 40-469 easily distinguished teams without any conflicts.

... So why then is this such a big problem for major sports leagues?

The reason most sports leagues have issues in not a lack of available combinations, but because some colors are just a lot more popular than others. For example: despite all of the colors to choose from, 88% of the teams in the NFL use black, blue, or red as one of their colors. 40% of the teams use 2 of these colors and 1 team uses all 3. In contrast, colors like teal, brown, and burgundy are only used by 1 team each and not a single team uses pink despite it being such a popular and recognizable color in other contexts. With this much overlap, it can get hard to tell who is who not because there is not enough colors, but because there is a strong cultural bias that causes most of the teams to gravitate towards these same 3 colors.

This is because in Western culture, black is highly associated with masculinity, red with aggression, and blue with comradery: The 3 principle virtues of the male athlete. In contrast: pink and teal are typically seen as more feminine colors and brown and burgundy are humble colors which goes against the typical virtues of an athlete.

In other words, if you want to get all of your teams to look different, it has to be through acts of regulation that force them to pick from an even distributions of colors instead of letting the teams pick their own colors. So, if you let the league assign colors instead of letting teams pick them, you could easily have 21 teams that each look uniquely deferent without needing separate home/away uniforms or other secondary features to distinguish them.

It's easy. Just use whatever colour you want so long as you have 11 distinct jerseys. Some plain, some striped, it doesn't matter. Then do the rest by changing the shorts/skirt colours.

You might have a grey and white striped jersey thats similar to a blue and white one. But the red shorts, versus the green tell you which is which at a glance.

Main colors are red, orange, yellow, green, blue, purple. Black and white are not considered technically colors but they are distinct.

So that makes about 8 main and distinct colors.

So some teams might have both shirts and pants the same color. That makes eight teams.

Other teams might use different colors for shirts and pants. Eight times eight is sixty four. If a team with red shirts and purple pants and a team with purple shirts and red pants are counted as having different color combination then there are one hundred twenty eight two color combinations in addition to the eight color combinations.

In heraldry white and yellow are counted as the metals silver and gold, and red, blue, green, purple, and black are counted as the colors. Two colors or two metals can be placed side by side in a pattern, but a color can not be placed upon a color, nor a metal upon a metal, to ensure sufficient contrast.

Thus using heraldic colors there would be ten color combinations, or twenty if a black shirt and gold pants are considered different from a gold shirt and black pants. And that is with the shirts and pants being the same color all over instead of having patterns or having images in a different color on them.

In medieval warfare being able to recognize the heraldic designs on shields, surcoats, and banners at a distance to tell friend from foe could be a matter of life and death, so if the members of team wear their coat of arms or their heraldic badge in a large size on their clothing it should be easy enough to tell who is who.

If the sports players and fans know the rules of whatever heraldry like system your society uses, Someone could described the heraldry of a team in a sentence or two and the people who hear that description will be able to recognize it at first sight from the description.

There is a figurative expression "draped in the flag" for someone who claims to be so patriotic that you can picture them wearing the flag wrapped around them. (Which can be literally true if someone has a coat of arms and banner of that coat of arms and wears clothing with the coat of arms on the clothing, the same pattern as on their banner.) And of course it is entirely possible for sports teams in our world to have their flags. Thus fictional sports teams might have their flags, and the uniforms of players might have their flags printed on them.

Solved in real world

When a sporting event includes multiple teams simultaneously, their attires have to all be different:

(Morio, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons)

The use of patterns, sponsor names and logos distinguishes the teams well enough for a profitable sport.

There are 10 teams in Formula 1 and 17 in NASCAR, close to your parameters.