First, you can read a short presentation of my results (abstract.pdf).
The thesis itself contains more background, more details and interesting theorems with their nice proofs.
Thus for a real abstract, see the preface in the thesis.
You can download whole thesis ( all.pdf), or separated chapters:
- Title pages (mff_intro.pdf) -
title page, acknowledgement, abstracts, preface
- Ramsey-type questions (ramsey_segments.pdf) -
Dilworth's theorem, partial ordering on segments, ramsey type question in segments, note on other ramsey type questions
- Geometric and Topological graphs (geom_top_graphs.pdf) -
plane graphs (Euler formula, Fary theorem, Hanani-Tutte theorem),
crossing numbers (Crossing lemma, Crossing lemma for graphs with girth>2r, embedding method, decay of crossing number),
different definitions of crossing number,
graphs with no forbidden subgraphs (k pairwise disjoint segments, k pairwise crossing segments, k crossing on an edge).
- Convex Independent Sets (convex.pdf) -
Erdos-Szekeres theorem (classical proofs using Ramsey theorem, proof using cups and caps),
cups and caps, note on projective mappings,
empty convex independent sets,
open cups and open caps.
- Circle Graphs (circle_graphs.pdf) -
coloring circle graphs,
lower bounds on coloring,
circle graphs with bounded clique and independent set(ω(G)<=k, α(G)<=l).
I will be happy, if the texts help you. If you have some comments, new suggestions, let me know.
Katedra Aplikované Matematiky (KAM),
Malá Strana, 3. patro, místnost 322
e-mail: kuba at matfyz.cz