Some sharp lower bounds for the bipartite Tur\'{a}n number of theta graphs

TL;DR

This paper extends algebraic construction techniques to establish new lower bounds on bipartite Turán numbers for theta graphs, revealing the maximum edges in bipartite graphs avoiding certain theta subgraphs.

Contribution

It generalizes Conlon's algebraic method to derive sharp lower bounds for bipartite Turán numbers of theta graphs for all odd k ≥ 3.

Findings

Established lower bounds for ex(n,m,θ_{k,c_k}) for odd k ≥ 3

Extended algebraic construction techniques to bipartite Turán problems

Provided explicit bounds involving parameters n, a, and k

Abstract

We expand Conlon's random algebraic construction to show that for any odd number $k \geq 3$ exists a natural number $c_k$ (the same as Conlon's) such that $\operatorname{ex}(n^a,n,\theta_{k,c_k}) = \Omega_{k,a}((n^{1 + a})^{\frac{k + 1}{2k}})$, with $a \in [\frac{k - 1}{k + 1}, 1)$. Where given a graph $H$, we denote by $\operatorname{ex}(n,m,H)$ the maximum number of edges an $H-$free bipartite graph can have when the cardinalities of its parts are $n$ and $m$. Also, we denote with $\theta_{k,l}$ the graph where two vertices are connected through $l$ disjoint paths of length $k$.