After ETH: the kxk Clique Hypothesis
Year: 2026
Type: article
Abstract: Two preprints from October 2026, by OpenAI and by Anthropic, claim to refute the Exponential Time Hypothesis (ETH). Weak ETH, the proposed fallback, is too weak for the known proofs of the slightly superexponential lower bounds 2^o(k log k) and of the lower bound f(k)·n^o(k) for Clique. This note proposes the k×k Clique Hypothesis: k×k Clique has no algorithm with running time 2^o(k log k). ETH implies it, so assuming it is no stronger than assuming ETH. Together with Weak ETH and two hypotheses that the preprints do not refute, it yields 20 of the 29 lower bounds in the ETH chapter of the textbook by Cygan et al. verbatim and the other 9 in weaker form. As far as we can see, the theorems of the preprints do not give time 2^o(k log k) for k×k Clique unless few pairs of rows are constrained, that is, have a non-edge between them. On such sparse instances, OpenAI's Theorem D.1 gives this time if it is correct.
Author: Szeider, Stefan
Institution: TU Wien
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224875827