The vertex Folkman number $F_v(3,3;5)$ equals~$8$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914551015931904 |
|---|---|
| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05526 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The vertex Folkman number $F_v(3,3;5)$ equals~$8$ Niu, Tong Combinatorics 05C55, 05C15, 05C69, 68R10 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. |
| title | The vertex Folkman number $F_v(3,3;5)$ equals~$8$ |
| topic | Combinatorics 05C55, 05C15, 05C69, 68R10 |
| url | https://arxiv.org/abs/2605.05526 |