On the independence number of sparser random Cayley graphs
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| Format: | Preprint |
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2024
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| author | Campos, Marcelo Dahia, Gabriel Marciano, João Pedro |
| author_facet | Campos, Marcelo Dahia, Gabriel Marciano, João Pedro |
| contents | The Cayley sum graph $Γ_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $\mathbb{Z}_n$ with $p = 1/2$, then the independence number of $Γ_A$ is asymptotically equal to $α(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$α(Γ_A) = (1 + o(1)) α(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$.
One of the tools in our proof is a geometric-flavoured theorem that generalises Freĭman's lemma, the classical lower bound on the size of high dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the independence number of sparser random Cayley graphs Campos, Marcelo Dahia, Gabriel Marciano, João Pedro Combinatorics Number Theory 11P70, 60C05, 05C80, 52A20 The Cayley sum graph $Γ_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $\mathbb{Z}_n$ with $p = 1/2$, then the independence number of $Γ_A$ is asymptotically equal to $α(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$α(Γ_A) = (1 + o(1)) α(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$. One of the tools in our proof is a geometric-flavoured theorem that generalises Freĭman's lemma, the classical lower bound on the size of high dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant. |
| title | On the independence number of sparser random Cayley graphs |
| topic | Combinatorics Number Theory 11P70, 60C05, 05C80, 52A20 |
| url | https://arxiv.org/abs/2406.09361 |