## Structures cannot be avoided! — Ramsey theory on the intersection graphs of line segments —

27 November 2017

This column is also available in pdf: filename RamseyLineSegment.pdf

Consider n line segments S1,,Sn on the plane in general position (ie., segments are either disjoint or intersect in exactly one point). We ask the following question:
How many pairwise disjoint segments OR how many pairwise intersecting segments are there?

Define the intersection graph G = (V,E) where each segment Si is associated to a corresponding node V i, and where there is an edge {V i,V j} if and only if the corresponding line segments intersect (Si Sj = ). A subset of pairwise intersecting segments corresponds to a clique in G, and a subset of pairwise disjoint segments corresponds to an independent set in G (an anti-clique also called a stable). Consider the complement graph G = (V,E) where E = \E, with = {{V i,V j}  :  i = j} the full edge set. G is the disjointness graph [3] of the segments (i.e., an edge between nodes if and only if corresponding segments are disjoint), and we have G G = Kn, the clique of size n.

Ramsey-type theorems are characterizing the following types of questions: “How large a structure must be to guarantee a given property?” Surprisingly, complete disorder is impossible! That is, there always exists (some) order in structures!

Define the Ramsey number R(s,t) as the minimum number n such that any graph with |V | = n nodes contains either an independent set of size s or a clique Kt of size t.

One can prove that those Ramsey numbers are all finite [4] (by proving the recursive formula R(s,t) R(s - 1,t) + R(s,t - 1) with terminal cases R(s,1) = R(1,t) = 1 for s,t 1), and that the following bound holds (due to the theorem of Erdös-Szekeres [2]):

When s = t, = is a central binomial coefficient that is upper bounded by 22s. Therefore the diagonal Ramsey number R(s) = R(s,s) is upper bounded by 22s. Furthermore, we have the following lower bound: 2 < R(s) when s [2]. Thus s ≥⌊ log 2n. Erdös proved using a probabilistic argument [1] that there exists a graph G such that s 2log 2n (G and G do not contain Ks subgraphs). It is proved in [3] (2017) a much stronger result that s = Ω(n) for intersection graphs of line segments: Thus there are always Ω(n) pairwise disjoint or pairwise intersecting segments in a set of n segments in general position.

In general, one can consider a coloring of the edges of the clique Kn into c colors, and asks for the largest monochromatic clique Km in the edge-colored Kn. For the pairwise disjoint/intersecting line segments, we have c = 2: Say, we color edges red when their corresponding segments intersect and blue, otherwise.

Ramsey’s theorem [4] (1930) states that for all c, there exists n m 2 such that every c-coloring of Kn has a monochromatic clique Km.

Let us conclude with the theorem on acquaintances (people who already met) and strangers (people who meet for the first time): In a group of six people, either at least three of them are pairwise mutual strangers or at least three of them are pairwise mutual acquaintances. Consider K6 (n = 6, 15 edges), and color an edge in red if the edge people already met and in blue, otherwise. Then there is a monochromatic triangle (m = 3). Proof: R(3) = R(3,3) = 6.

Well, it is known that R(4) = 18 but R(5) is not known! We only know that 43 R(5) 48, 102 R(6,6) 165, etc. Quantum computers [5] can be used to compute Ramsey numbers!

Initially created 27th November 2017 (last updated November 30, 2017).

### References

[1]   Paul Erdös. Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4):292–294, 1947.

[2]   Paul Erdös and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463–470, 1935.

[3]   J. Pach, G. Tardos, and G. Toth. Disjointness graphs of segments. ArXiv e-prints, April 2017.

[4]   Frank P Ramsey. On a problem of formal logic. In Classic Papers in Combinatorics, pages 1–24. Springer, 2009.

[5]   Hefeng Wang. Determining Ramsey numbers on a quantum computer. Physical Review A, 93(3), 2016.