General Information
| Student: | Peter Korcsok |
|---|---|
| Office: | 444 |
| School: | Charles University in Prague |
| E-mail: | peter.korcsok guess_what_is_here rutgers.edu |
| Projects: | Binary Kakeya Sets |
| $L(p,q)$-labeling of Interval Graphs |
Binary Kakeya Sets
Project Description
For $x,y \in \{0,1\}^n$ we define the line $L_{x,y} = \{x^i \oplus y \mid i = 0, ..., n-1\}$ of direction $x$ where $x^i$ denotes the $i$-th rotation of $x$. We say that a set $S \subseteq \{0,1\}^n$ is a Kakeya set if for every $x \in \{0,1\}^n$ there exists $y \in \{0,1\}^n$ such that $L_{x,y} \subseteq S$. Michal Koucký asks for the smallest Kakeya set for a given $n$.Previous work
- J. Bourgain Harmonic analysis and combinatorics: How much may they contribute to each other? IMU/Amer. Math. Soc., pages 13–32, 2000.
- Zeev Dvir On the size of Kakeya sets in finite fields, arXiv:0803.2336 [math.CO].
Current activities
First week (June 2nd - June 6th)
- Introduction to the problem.
- Reading the article by Zeev Dvir.
- Few first observations.
- Computer program for finding all Kakeya sets for dimensions up to 5.
Second week (June 9th - June 13th)
- Slightly better upper bound.
- Computer program for finding out size of Kakeya sets for dimension 6. (Program failed because of using too much memory.)
- Some computer experiments on sets from the first week.
- Some small observations.
Third week (June 16th - June 20th)
- Computer program for finding out size of Kakeya sets for dimension 6 using external SAT solver. (Program killed because it would run too long.)
- Integer program for finding out size of Kakeya sets for dimension 6.
- Integer program for finding out size of Kakeya sets for dimension 7. (Program failed because of using too much memory.)
- More small observations.
Fourth week (June 23th - June 27th)
- Nontrivial lower bound - roughly $2^{n/2}$.
- Observation about "surronded" point - point whose neighborhood is subset of Kakeya set.
- Observation that each Kakeya set can be "normalized" - transformed in such way that it contains all unit vectors.
- Computer program for finding all normalized Kakeya sets for dimensions up to 5.
- Preparations for Cultural Day.
Fifth week (June 30th - July 4th)
- Computer program for finding all normalized Kakeya sets for dimension 6. (Program killed because it would run too long.)
- Some experiments on normalized sets from the previous week.
- Some improvements of the program for finding all normalized Kakeya sets for dimension 6. (Program is still slightly inefficient and still running at the end of the week.)
- Some partial results for finding all normalized Kakeya sets for dimension 6.
Sixth week (July 7th - July 11th)
Week has not finished yet.Seventh week (July 14th - July 18th)
Week has not started yet.Coworkers
Back to top$L(p,q)$-labeling of Interval Graphs
Project Description
Interval graphs are intersection graphs of a family of intervals of real numbers. $L(p,q)$-labeling of a graph $G$ is a mapping $l \colon V_G \to X$ where $X \subset \mathbb{Z}$ such that- $|f(u) - f(v)| \geq p$ whenever the vertices $u$ and $v$ are connected by an edge,
- $|f(u) - f(v)| \geq q$ whenever there exists some vertex $w$ such that both $u$ and $v$ are neighbors of $w$.
Previous work
- Peter Che Bor Lam, Guohua Gu, Wai Chee Shiu, Tao-Ming Wang: On Distance Two Labelling of Unit Interval Graphs (see the paper).
- G.J. Chang; D. Kuo, The L(2,1)-labeling problem on graphs. SIAM J. Discrete Math. 9 (1996), 309--316.
- D. Sakai, Labelling chordal graphs: distance two condition. SIAM J. Disc. Math. 7 (1994), 133--140.
- J.R. Griggs; R.K. Yeh, Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5 (1992), 586--595.
Current activities
See progress on Veronica's webpage.Coworkers
Back to topPresentations
- First Presentation (A brief introduction to intersection graphs)
- Culural Day Presentation (Slovakia)
Additional Information
My Mentorhttp://www.mgvis.com/
http://www.mgvis.com/AbelloVitaResearchOct08.html