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arXiv · 2405.02864

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

Stefanos Theodorakopoulos
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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 $k$.

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Stefanos Theodorakopoulos. 2024-05-05. Some sharp lower bounds for the bipartite Tur\'{a}n number of theta graphs. https://arxiv.org/abs/2405.02864

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