---

Claim. Let $C_k$ be the connective constant of Problem 528. S. E. Alm, Upper bounds for the connective constant of self-avoiding walks, gives a method that bounds the connective constant of a lattice above by the largest eigenvalue of a matrix indexed by short self-avoiding walks, for a class of lattices that includes every lattice studied in connection with self-avoiding walks, and applies it numerically. For the square lattice it proves

$$C_2\le2.696,$$

the bound as the paper's abstract states it (printed there as $\mu<2.696$); the abstract also states $\mu<4.278$ for the triangular lattice and $\mu<4.756$ for the simple cubic lattice, the latter being $C_3<4.756$ in the problem's notation. Together with [[problems/discrete_geometry/E0528/claims/1993_08_07_conway_guttmann|Conway and Guttmann's lower bound]] this places $C_2$ in $[2.62,2.696]$.

Covers. The upper bound $C_2\le2.696$, and the upper bound $C_3<4.756$ that the same paper states. Not covered: any lower bound, any $k\ge4$, and the value of $C_k$ for any $k$, which is not determined.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: S. E. Alm, Upper bounds for the connective constant of self-avoiding walks, Combin. Probab. Comput. 2 (1993), no. 2, 115--136. The site's commentary records the square-lattice bound, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence. The proof is not compiled in this corpus.

Dating. The page is dated by the issue month in the publisher's record, June 1993; the day is a placeholder.