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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.02864 |
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| _version_ | 1866910577451859968 |
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| author | Theodorakopoulos, Stefanos |
| author_facet | Theodorakopoulos, Stefanos |
| contents | 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 $\operatorname{ex}(n^a,n,θ_{k,c_k}) = Ω_{k,a}((n^{1 + a})^{\frac{k + 1}{2k}})$, 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 $θ_{k,l}$ the graph where two vertices are connected through $l$ disjoint paths of length $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some sharp lower bounds for the bipartite Turán number of theta graphs Theodorakopoulos, Stefanos Combinatorics 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 $\operatorname{ex}(n^a,n,θ_{k,c_k}) = Ω_{k,a}((n^{1 + a})^{\frac{k + 1}{2k}})$, 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 $θ_{k,l}$ the graph where two vertices are connected through $l$ disjoint paths of length $k$. |
| title | Some sharp lower bounds for the bipartite Turán number of theta graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.02864 |