Large line-free sets and their applications
Jakob Führer, Vladislav Taranchuk
TL;DR
The paper develops explicit polynomial-based constructions over finite fields to produce large line-evasive sets, enabling a partition of $\mathbb{F}_q^n$ into such sets with size $q^{n(1-\frac{2}{t^2+t})}$. These constructions yield new lower bounds for vector space Ramsey numbers $R_q(2;k)$, explicit colorings with $\chi_q(\binom{q}{2}+1)\le q$, and improved asymptotics for $ex(n,m,\{C_4,\theta_{3,t}\})$, including $ex(n,n^{2/3},\{C_4,\theta_{3,3}\})=\Theta(n^{1+1/9})$. The methods extend to projective spaces via field-reduction, delivering direct consequences for vector space Ramsey numbers in PG$(n,q)$ with $q$- and $k$-dependence. Overall, the work advances extremal combinatorics through explicit, maximal $t$-line evasive constructions and demonstrates broad impact on both Ramsey-type problems and Turán-type extremal bounds.
Abstract
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.