The Triangle Game, from Zero
All 120 connections between sixteen people receiving one of three colours without making a one-colour triangle<br>The animation shows sixteen people with a connection between every pair. That makes 120 connections, and each one receives one of three colours. When the colouring is complete, there is no group of three people whose three mutual connections all have the same colour. Sixteen is the largest group for which this is possible with three colours. Greenwood and Gleason built this example and proved that seventeen is impossible in 1955. Everything in this guide comes from the open source repository at github.com/muchmirul/conjectures: the text, every figure, and the tests behind each number.
This guide asks what happens when many more colours are available. More colours let us keep a larger group safe, but how much larger can the group become for each added colour? Erdős offered cash prizes for answering that question. For decades, the best construction and the best limit were so far apart that researchers did not even know whether the long-term rate stopped at a fixed number. A result in 2026 showed that it never stops growing.
The guide develops the result in small steps. Most numbered sections also have a page you can play with, where you can change the main choices and watch the picture respond.
The code that creates every figure is included in this repository. The tests recalculate the numbers and check the small colourings shown here. When a statement comes from an existing theorem rather than those tests, the text says so.
The 2026 work is presented in two documents. One gives the finished proof. The other is a behind-the-scenes guide that explains how the ideas were found, including earlier attempts that failed. Sections 5 to 7 explain why the old methods stalled. Sections 8 to 11 explain the finished construction. A table near the end shows where each idea appears in the two source documents.
first, learn the game 1 the rule<br>2 why six people cannot stay safe<br>3 what an extra colour changes<br>then, find the question 4 how to compare different colour counts<br>why earlier ideas stalled 5 multiplying safe groups<br>6 a general upper limit<br>7 the wide gap between them<br>the finished construction 8 which colours each room omits<br>9 keeping arrivals on safe teams<br>10 one fixed referee for every choice<br>11 stacking the rooms into a tower<br>what the result settles 12 what the result settles<br>Use the controls to change the picture. Open the activity on its own page
1 · The game
Start with a group of people and draw a connection between every pair. Give every connection a colour. Any choice of three people forms a triangle because each pair among them is connected. You lose if all three connections in one of those triangles have the same colour.
We will call this a one-colour triangle . A completed colouring with no one-colour triangle will be called safe . A group together with a safe colouring is a safe table .
Two colourings of four people, with a one-colour triangle on the left and a safe result on the right<br>The thick triangle on the left uses one colour on all three sides, so that colouring loses. Four people contain four possible groups of three. On the right, each of those four triangles uses at least two colours, so the colouring is safe.
With two colours, a group of five can be kept safe. Apart from renaming the people or swapping the colours, there is only one way to do it. Place the five people around a circle. Use the first colour for neighbouring pairs around the outside, then use the second colour for the five connections that skip across the circle.
The ten connections among five people being coloured, followed by a check of all ten triangles<br>The final sweep checks every possible group of three. There are ten of them, and every one uses both colours. Safety is therefore not based on how the picture looks. It comes from checking a short, complete list. The tests in this repository perform the same check.
There is also a quick way to understand the pattern. Look at either colour by itself. Its five connections form a ring, and a ring contains no triangle. Since neither colour can make a triangle on its own, the combined colouring is safe.
Use the controls to change the picture. Open the activity on its own page<br>2 · Six is forced
Five people can stay safe with two colours. Six people cannot. This is the one theorem proved completely in this guide, and every step can be seen in the animation.
Choose one of the six people and look at the five connections leading from that person. There are only two colours available. At least three of those five connections must have the same colour, because two connections of each colour would account for only four. Suppose three of them are red.
Three same-coloured connections from one person forcing a red or blue triangle<br>Now focus on the three people at the other ends of those red connections. If any pair among them is joined in...