Problem 528

Erdős Problem #528: declared status 'open'. Formalized: no. Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine$$C_k=\lim_{n\to\infty}f(n,k)^{1/n}.$$ Current best: Hammersley and Morton [HM54] showed that this limit exists, and it is trivial that $k\leq C_k\leq 2k-1$. Conway and Guttmann [CG93] showed that $C_2\geq 2.62$ and Alm [Al93] showed that $C_2\leq 2.696$. J., Lower bound on the connective constant for square lattice self-avoiding walks. Prize: no. OEIS: A156816, A387897. Tags: geometry.