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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1812.08396v1">arXiv:1812.08396v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eblbosvmcvdz3j765y5g6kmgym">fatcat:eblbosvmcvdz3j765y5g6kmgym</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191013040148/https://arxiv.org/pdf/1812.08396v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b9/aa/b9aa575cf7fd7590e4f5350038539b03aa156c07.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1812.08396v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>