Chromatic number in 1.9999^n time? Fast deterministic set partitioning under the asymptotic rank conjecture

In this paper we further explore the recently discovered connection by Björklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an n-vertex graph can be computed deterministically in O(1.99982^n) time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the 2^npoly(n) time algorithm for chromatic number by Björklund, Husfeldt, and Koivisto [SICOMP 2009]. Our technique is a combination of earlier algorithms for detecting k-colorings for small k and enumerating k-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case.