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arXiv · 2606.02210

Vertex-colored Tur\'{a}n theorems with applications in extremal hypergraph problems

Wanfang Chen·Jinghua Deng·Jianfeng Hou·Xizhi Liu·Yixiao Zhang
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Abstract

Balogh, Clemen, and Lidick\'{y} proved that the $\ell_{2}$-norm Tur\'{a}n 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 Tur\'{a}n-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.

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Wanfang Chen, Jinghua Deng, Jianfeng Hou, Xizhi Liu, Yixiao Zhang. 2026-06-01. Vertex-colored Tur\'{a}n theorems with applications in extremal hypergraph problems. https://arxiv.org/abs/2606.02210

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