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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.18418 |
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| _version_ | 1866917966831943680 |
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| author | Düzgün, Baran Riet, Ago-Erik Taranchuk, Vladislav |
| author_facet | Düzgün, Baran Riet, Ago-Erik Taranchuk, Vladislav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New constructions of unbalanced $\{C_4,θ_{3, t}\}$-free bipartite graphs Düzgün, Baran Riet, Ago-Erik Taranchuk, Vladislav Combinatorics 05C35 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. |
| title | New constructions of unbalanced $\{C_4,θ_{3, t}\}$-free bipartite graphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2503.18418 |