تقرير
On the independence number of sparser random Cayley graphs
| العنوان: | On the independence number of sparser random Cayley graphs |
|---|---|
| المؤلفون: | Campos, Marcelo, Dahia, Gabriel, Marciano, João Pedro |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Number Theory |
| الوصف: | The Cayley sum graph $\Gamma_A$ 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 $\Gamma_A$ is asymptotically equal to $\alpha(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$\alpha(\Gamma_A) = (1 + o(1)) \alpha(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$. One of the tools in our proof is a geometric-flavoured theorem that generalises Fre\u{i}man'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. Comment: Fixed problems spotted by Zach Hunter, moved proof of Chang's theorem in Z/nZ to another appendix. 44 pages + 8 page appendix |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.09361 |
| رقم الأكسشن: | edsarx.2406.09361 |
| قاعدة البيانات: | arXiv |
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