Vertex-colored Turán theorems with applications in extremal hypergraph problems
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914622977605632 |
|---|---|
| author | Chen, Wanfang Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhang, Yixiao |
| author_facet | Chen, Wanfang Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhang, Yixiao |
| contents | Balogh, Clemen, and Lidický proved that the $\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard.
The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02210 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex-colored Turán theorems with applications in extremal hypergraph problems Chen, Wanfang Deng, Jinghua Hou, Jianfeng Liu, Xizhi Zhang, Yixiao Combinatorics Balogh, Clemen, and Lidický proved that the $\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction. |
| title | Vertex-colored Turán theorems with applications in extremal hypergraph problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.02210 |