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

We expand Conlon's random algebraic construction to show that for any odd number k ≥ 3 exists a natural number cₖ (the same as Conlon's) such that ex(nᵃ,n,θk,cₖ) = Ωk,a((n1 + a)k + 1/2k), with a ∈ [k - 1/k + 1, 1). Where given a graph H, we denote by 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 θk,l the graph where two vertices are connected through l disjoint paths of length k.