Czipszer proved that $g_k(n)$ exists for all $k$, and in fact $g_k(n)\leq (k+1)n$. Erdős wrote it is 'easy to see' that\[g_k(n)\geq (k+1)n-(k+1)^2.\]Pósa proved that $g_1(n)=2n-4$ for $n\geq 4$. Erdős could prove the conjectured equality for $n\geq 2k+2$ when $k=2$ or $k=3$.

The conjectured equality was proved for $n\geq 3k+3$ by Jiang [Ji04].

Curiously, in [Er69b] Erdős mentions this problem, but states that his conjectured equality for $g_k(n)$ was disproved (for general $k$) by Lewin, citing oral communication. Perhaps Lewin only disproved this for small $n$, or perhaps Lewin's disproof was simply incorrect.

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This page was last edited 06 October 2025.