Math 2.0

Combinatorics Theorem

Erdős theorem on graphs of large girth and large chromatic number

Builds on Chromatic numberGirthGraph

Statement

Let g≥3g \ge 3 and tt be integers. Then there is a graph GG whose chromatic number satisfies χ(G)>t\chi(G) > t and whose girth is greater than gg.

Remarks

So a graph can need many colours even though it has no short cycles at all. Large chromatic number does not force a small cycle, such as a triangle, into the graph. The classical proof uses the probabilistic method on a random graph.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionGraph colouring, definition, no evidence levelGraph colouringdefinitionWalk, definition, no evidence levelWalkdefinitionK-colourable graph, definition, no evidence levelK-colourable graphdefinitionTrail, definition, no evidence levelTraildefinitionChromatic number, definition, no evidence levelChromatic numberdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionGirth, definition, no evidence levelGirthdefinitionErdős theorem on graphs of large girth and large chromatic number, statement, no evidence levelErdős theorem on grap…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Graph colouring definition
  3. Walk definition
  4. K-colourable graph definition
  5. Trail definition
  6. Chromatic number definition
  7. Circuit definition
  8. Cycle definition
  9. Acyclic graph definition
  10. Girth definition
  11. Erdős theorem on graphs of large girth and large chromatic number statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_ramsey_girth.ptx#L12, source/s_ramsey_girth.ptx:12
About this page
Tag5SKPQNWNGD
URIm2://5SKPQNWNGD
Kindstatement, theorem
WritingW0 stub
MSC05C15, 05C38, 05C80
StatusThis copy of the site has no build data. Run m2 status 5SKPQNWNGD.
AboutChromatic number, Girth
UsesChromatic number, Girth, Graph
Changed4 November 2026

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