Clique covers of complete graphs and piercing multitrack intervals

TL;DR

This paper investigates the piercing properties of k-wise intersecting t-intervals and extends the results to vertex coverings in edge-colored complete graphs with specific induced subgraph restrictions, providing asymptotic bounds and new Ramsey-type problems.

Contribution

It establishes an asymptotic lower bound for piercing t points in k-wise intersecting t-intervals and explores vertex coverings in edge-colored graphs with induced C4-free color classes, linking geometric and graph-theoretic problems.

Findings

At least (k-1)/(k+1) fraction of k-wise intersecting t-intervals can be pierced by t points.

In edge-colored complete graphs with C4-free color classes, at least 4/5 of vertices can be covered by two monochromatic cliques.

Results generalize to subtrees of a tree and lead to new Ramsey-type problems.

Abstract

Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings.