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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2403.18611 |
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| _version_ | 1866915789240532992 |
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| author | Führer, Jakob Taranchuk, Vladislav |
| author_facet | Führer, Jakob Taranchuk, Vladislav |
| contents | In this paper, we construct explicit families of polynomials $P \in \mathbb{F}_q[x_1,\dots,x_n]$ with large root sets which have restricted intersections with affine lines. We use these sets to make substantial progress on a number of problems in extremal combinatorics.
For each prime power $q$ and integer $2 \le t \le q-1$, we construct $t$-line evasive subsets of $\mathbb{F}_q^n$ of size \[ q^{\,n\left(1-\frac{2}{t^2+t}\right)}, \] which is significantly larger than those previously known. Moreover, our method yields a partition of $\mathbb{F}_q^n$ into such sets.
We extend this partitioning result to the projective space $PG(n,q)$, obtaining the first explicit colorings for the vector space Ramsey number $R_q(2;k)$ that exhibit dependence on both $q$ and $k$. In particular, we show that \[ R_q(2;k) > \frac{(q-1)k}{2} - O_q(1), \] improving recent bounds.
Finally, we apply these constructions to extremal graph theory and improve the best-known bounds on the bipartite Turán number $ \mathrm{ex}(n,m,\{C_4,θ_{3,t}\})$. Most notably, we show that \[ \mathrm{ex}(n,n^{2/3},\{C_4,θ_{3,3}\}) = Θ(n^{1+1/9}), \] making progress on a question originally posed by Erdős. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large line-free sets and their applications Führer, Jakob Taranchuk, Vladislav Combinatorics In this paper, we construct explicit families of polynomials $P \in \mathbb{F}_q[x_1,\dots,x_n]$ with large root sets which have restricted intersections with affine lines. We use these sets to make substantial progress on a number of problems in extremal combinatorics. For each prime power $q$ and integer $2 \le t \le q-1$, we construct $t$-line evasive subsets of $\mathbb{F}_q^n$ of size \[ q^{\,n\left(1-\frac{2}{t^2+t}\right)}, \] which is significantly larger than those previously known. Moreover, our method yields a partition of $\mathbb{F}_q^n$ into such sets. We extend this partitioning result to the projective space $PG(n,q)$, obtaining the first explicit colorings for the vector space Ramsey number $R_q(2;k)$ that exhibit dependence on both $q$ and $k$. In particular, we show that \[ R_q(2;k) > \frac{(q-1)k}{2} - O_q(1), \] improving recent bounds. Finally, we apply these constructions to extremal graph theory and improve the best-known bounds on the bipartite Turán number $ \mathrm{ex}(n,m,\{C_4,θ_{3,t}\})$. Most notably, we show that \[ \mathrm{ex}(n,n^{2/3},\{C_4,θ_{3,3}\}) = Θ(n^{1+1/9}), \] making progress on a question originally posed by Erdős. |
| title | Large line-free sets and their applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.18611 |