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The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs

Theorem 1.1: for every r≥4r\ge4 there is cr>0c_r>0 such that every finite KrK_r-free graph with nn vertices and average degree d≥2d\ge2 has α(G)≥crnlog⁡d/d\alpha(G)\ge c_r n\log d/d, removing the log⁡log⁡d\log\log d loss in Shearer's 1995 bound. The proof maximizes an entropy functional over vertex weights and shows that the weighted triangle mass is at most OrO_r times the edge mass, via random multipliers, vertex splitting and a recursion on the clique size. The constant crc_r is not optimized and no leading coefficient (as in Shearer's triangle-free result) is claimed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs · AEKS independence conjecture · Problem 802

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Artifacts and verifiers

Lean: OAI/Combinatorics/CliqueFree/Main.lean (logarithmic_independence_bound)

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

Erdos Problem #802 evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean: OAI/Combinatorics/CliqueFree/Main.lean (logarithmic_independence_bound) evidence for this event

A logarithmic independence bound for clique-free graphs (companion) evidence for this event

VibeMathed record: The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs evidence for this event

The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs parent of this event

Theorem 1.1: for every r≥4r\ge4 there is cr>0c_r>0 such that every finite KrK_r-free graph with nn vertices and average degree d≥2d\ge2 has α(G)≥crnlog⁡d/d\alpha(G)\ge c_r n\log d/d, removing the log⁡log⁡d\log\log d loss in Shearer's 1995 bound. The proof maximizes an entropy functional over vertex weights and shows that the weighted triangle mass is at most OrO_r times the edge mass, via random multipliers, vertex splitting and a recursion on the clique size. The constant crc_r is not optimized and no leading coefficient (as in Shearer's triangle-free result) is claimed. parent of this event

The Alon-Krivelevich-Sudakov conjecture: graphs with a fixed forbidden subgraph have chromatic number O(Delta/log Delta) evidence for this event

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