The largest $K_r$-free set of vertices in a random graph
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| Format: | Preprint |
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2026
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| _version_ | 1866912970939826176 |
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| author | Bohman, Tom Michelen, Marcus Mubayi, Dhruv |
| author_facet | Bohman, Tom Michelen, Marcus Mubayi, Dhruv |
| contents | For $r \ge 2$ and a graph $G$, let $α_{r}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $α_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $α_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16454 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The largest $K_r$-free set of vertices in a random graph Bohman, Tom Michelen, Marcus Mubayi, Dhruv Combinatorics Probability For $r \ge 2$ and a graph $G$, let $α_{r}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $α_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $α_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$. |
| title | The largest $K_r$-free set of vertices in a random graph |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2603.16454 |