Vertex-colored Turán theorems with applications in extremal hypergraph problems
Abstract
Balogh, Clemen, and Lidický proved that the ℓ2-norm Turán problem for K53 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 K53-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 ℓ2-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced K4-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.
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BibTeX
@misc{chen2026vertexcolored,
title = {{Vertex-colored Turán theorems with applications in extremal hypergraph problems}},
author = {Wanfang Chen and Jinghua Deng and Jianfeng Hou and Xizhi Liu and Yixiao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.02210},
eprint = {2606.02210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02210},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vertex-colored Turán theorems with applications in extremal hypergraph problems
AU - Wanfang Chen
AU - Jinghua Deng
AU - Jianfeng Hou
AU - Xizhi Liu
AU - Yixiao Zhang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2606.02210
UR - https://arxiv.org/abs/2606.02210
AB - Balogh, Clemen, and Lidický proved that the ℓ_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 ℓ_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.
ER -