Result 184, Combinatorics

Correspondence coloring with a fixed forbidden subgraph

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need OF(Δ/log⁡Δ)O_F(\Delta/\log\Delta) colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size Ωr(nlog⁡d/d)\Omega_r(n\log d/d).

Lean formalization Proof

The bigger picture

Why it matters

A graph can have many connections while still avoiding a particular small pattern. These manuscripts report that such exclusions force more efficient coloring and, for forbidden cliques, large groups of mutually nonadjacent vertices.

What changes?

The coloring manuscript reports that excluding any fixed graph F as an ordinary subgraph gives a correspondence-coloring bound of C_F times Delta divided by log Delta, for sufficiently large maximum degree Delta. Here Delta is the largest number of neighbors of a vertex; C_F and the threshold depend only on F. Correspondence coloring allows edge-specific matchings of incompatible color choices. The bound also holds for ordinary coloring and list coloring, where vertices have prescribed color lists.

What does that help mathematicians do?

The companion manuscript reports that, for fixed integer r at least 4, an n-vertex graph without an r-vertex clique has an independent set of size at least c_r times n log d divided by d, provided its average degree d is at least 2. Here c_r is positive and depends only on r. Cliques are fully connected; independent sets have no internal edges. Using average rather than maximum degree allows unusually highly connected vertices.

Are there practical applications?

The immediate value is foundational: the coloring claim extends a degree-based guarantee to edge-specific incompatibilities, not just the rule that adjacent vertices must receive different colors. This strengthens what combinatorial researchers can deduce from an excluded subgraph. The abstracts state existence bounds, not an implemented coloring method or a demonstrated practical speedup.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Correspondence coloring graphs with a forbidden clique

October 5, 2026 29 pages

For every fixed integer r ≥ 4, we prove that every Kr-free graph of sufficiently large maximum degree Δ has correspondence chromatic number Or(Δ/log⁡Δ)O_r(\Delta/\log\Delta). This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026,
  author = {{OpenAI}},
  title = {{Correspondence coloring graphs with a forbidden clique}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf}{OAI:Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026}},
  year = {2026}
}

A logarithmic independence bound for clique-free graphs

September 25, 2026 21 pages Main result formalized in Lean

For every fixed integer r ≥ 4, every Kr-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least crnlog⁡d/dc_r n\log d/d, where cr>0c_r\gt 0 depends only on r. This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026,
  author = {{OpenAI}},
  title = {{A logarithmic independence bound for clique-free graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf}{OAI:A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/184.md.

Correspondence coloring with a fixed forbidden subgraph

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result gives a logarithmic improvement in the independence number of clique-free graphs. For every fixed integer r≥4r\ge4, there is cr>0c_r>0 such that every finite KrK_r-free simple graph on nn vertices with average degree d≥2d\ge2 has an independent set of size at least crnlog⁡d/dc_r n\log d/d. The constant depends only on rr.

Comparator links

Result Comparator statement
Logarithmic independence bound for clique-free graphs CliqueFreeLog.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.