In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of Kt,t, then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

中文翻译：

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框相交图的 Turán 型结果

在这篇简短的笔记中，我们证明了以下 Kővári-Sós-Turán 定理的类比，用于盒子的交集图。如果G是的交集图n轴平行的盒子在$${{\mathbb{R}}^d}$$这样G不包含副本ķt,t， 然后G最多有ctn（ 日志n)2d+3边缘，其中C=C(d)>0 仅取决于d. 我们的证明是基于探索boxicity、分离维度和poset维度之间的联系。使用这种方法，我们还证明了 Basit、Chernikov、Starchenko、Tao 和 Tran 的构造ķ2,2- 平面中点和矩形的无关联图可用于反驳 Alon、Basavaraju、Chandran、Mathew 和 Rajendraprasad 的猜想。我们表明存在具有超线性边数的分离维数 4 的图。