New constructions of unbalanced $\{C_4,θ_{3, t}\}$-free bipartite graphs

Abstract: In 1979, Erd\H{o}s conjectured that if $m = O(n^{2/3})$, then $ex(n, m, {C_4, C_6 }) = O(n)$. This conjecture was disproven by several papers and the current best-known bounds for this problem are $$ c_1n^{1 + \frac{1}{15}} \leq ex(n, n^{2/3}, {C_4, C_6}) \leq c_2n^{1 + 1/9} $$ for some constants $c_1, c_2$. A consequence of our work here proves that $$ ex(n, n^{2/3}, { C_4, \theta_{3, 4} }) = \Theta(n^{1 + 1/9}). $$ More generally, for each integer $t \geq 2$, we establish that $$ ex(n, n^{\frac{t+2}{2t+1}}, { C_4, \theta_{3, t} }) = \Theta(n^{1 + \frac{1}{2t+1}}) $$ by demonstrating that subsets of points $S \subseteq \text{PG}(n,q)$ for which no $t+1$ points lie on a line give rise to ${ C_4, \theta_{3, t} }$-free graphs, where PG$(n,q)$ is the projective space of dimension $n$ over the finite field of $q$ elements.