Large line-free sets and their applications

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 Tur\'an number $\mathrm{ex}(n,m,\{C_4,\theta_{3,t}\})$. Most notably, we show that $$\mathrm{ex}(n,n^{2/3},\{C_4,\theta_{3,3}\}) = \Theta(n^{1+1/9}),$$ making progress on a question originally posed by Erd\H{o}s.