### On Partitions of Two-Dimensional Discrete Boxes [article]

Eyal Ackerman, Rom Pinchasi
<span title="2018-12-20">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let A and B be finite sets and consider a partition of the discrete box A × B into sub-boxes of the form A' × B' where A' ⊂ A and B' ⊂ B. We say that such a partition has the (k,ℓ)-piercing property for positive integers k and ℓ if every line of the form {a}× B intersects at least k sub-boxes and every line of the form A ×{b} intersects at least ℓ sub-boxes. We show that a partition of A × B that has the (k, ℓ)-piercing property must consist of at least (k-1)+(ℓ-1)+ 2√((k-1)(ℓ-1)) sub-boxes.
more &raquo; ... s bound is nearly sharp (up to one additive unit) for every k and ℓ. As a corollary we get that the same bound holds for the minimum number of vertices of a graph whose edges can be colored red and blue such that every vertex is part of red k-clique and a blue ℓ-clique.
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