Math 2.0

Combinatorics Concept

Repair radii and villainy

Builds on Graph colouringDistance from a point to a set

Summary

A repair radius measures how far a colouring of a graph GG with a given profile of class sizes can be from a proper one.

The radius RG(m)R_G(m) is the largest Hamming distance from a point to a set, over colourings with profile mm, to the set of proper colourings with profile mm.

The weak radius RGw(m)R^w_G(m) lets the nearest proper colouring have any profile.

The villainy B(G)B(G) and the weak villainy Bw(G)B_w(G) are the maxima of these radii over feasible profiles, and both are zero for the empty graph.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionGraph colouring, definition, no evidence levelGraph colouringdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionInfimum, definition, no evidence levelInfimumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionDistance from a point to a set, definition, no evidence levelDistance from a point…definitionRepair radii and villainy, concept, no evidence levelRepair radii and vill…concept
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. Element definition
  2. Graph colouring definition
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Infimum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Distance from a point to a set definition
  20. Repair radii and villainy concept

Further reading

  1. mention ax://paper/2610.13556#L0DK, Definition 2.2. the paper that states the item (corpus arxiv AX9.2)

In the catalogue

About this page
TagEBNHD7FQ23
URIm2://EBNHD7FQ23
Kindconcept
WritingW1 stated
MSC05-XX
UsesGraph colouring, Distance from a point to a set
Changed3 November 2026

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What this page needs for W2

  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

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History

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DateChangeFieldBy
2026-11-02addthe whole nodetamnd

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