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arXiv · 2605.05526

The vertex Folkman number $F_v(3,3;5)$ equals~$8$

Tong Niu
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Abstract

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The witness is the cone $K_1 \vee \overline{C_7}$, a single universal vertex joined to the complement of a $7$-cycle. That this graph is $K_5$-free and arrows $(3,3)^v$ follows from a short independence-number argument. The matching lower bound -- no $K_5$-free graph on $7$ or fewer vertices works -- comes from exhaustive enumeration via nauty and a SAT check using Glucose\,4. The appendix has a self-contained Python script for verification.

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Tong Niu. 2026-05-07. The vertex Folkman number $F_v(3,3;5)$ equals~$8$. https://arxiv.org/abs/2605.05526

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