New constructions of unbalanced $\{C_4,θ_{3, t}\}$-free bipartite graphs
Baran Düzgün, Ago-Erik Riet, Vladislav Taranchuk
TL;DR
The paper investigates unbalanced bipartite graphs free of C4 and theta_{3,t} and establishes tight asymptotics ex(n, n^{(t+2)/(2t+1)}, {C4, theta_{3,t}}) = Θ(n^{1+1/(2t+1)}). It introduces linear representations of point sets in PG(n,q) and proves that selecting S with no t+1 collinear points yields {C4, theta_{3,t}}-free incidence graphs, enabling explicit lower bounds that match existing upper bounds. In particular, it shows ex(n, n^{2/3}, {C4, theta_{3,4}}) = Θ(n^{1+1/9}) and provides a general construction for all t, bridging finite geometry with extremal graph theory. The work suggests that the true value of ex(n, n^{2/3}, {C4, C6}) may be closer to current upper bounds and highlights the role of projective-space methods in deriving unbalanced Turán-type results.
Abstract
In 1979, Erdős 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 $$ ex(n, n^{2/3}, \{ C_4, θ_{3, 4} \}) = Θ(n^{1 + 1/9}). $$ More generally, for each integer $t \geq 2$, we establish that $$ ex(n, n^{\frac{t+2}{2t+1}}, \{ C_4, θ_{3, t} \}) = Θ(n^{1 + \frac{1}{2t+1}}) $$ 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 $\{ C_4, θ_{3, t} \}$-free graphs, where PG$(n,q)$ is the projective space of dimension $n$ over the finite field of $q$ elements.