## Project Description

I am part of a czech group and we will be working together on several projects - curve complexity of planar graphs, Erdős–Faber–Lovász conjecture and something with CPUs.

It seems that Aneta and I will mainly work on a project about a structure called tetris. It is about graph coloring and it is closely related to Erdős–Faber–Lovász conjecture (EFL). For further information about other projects see webpages of my colleagues.

Our goal is to read and traslate known results and proofs about EFL to tetris–language and possibly to make some improvements and new discoveries.

I also hope to meet new people and see a little of US, where I have never been to before.

## Progress

### First Week

June 1st - June 7th
• I arrived to Rutgers and met people who made REU happen. We have done the paperwork and other organization stuff, uff.
• I read few papers on EFL. I would like to improve the result about dense graphs.
• I bought a guitar. I rode a bike 12 miles to get it but it was worth it - it is making us happy.
• I also like Peter's and Michal's project about curve complexity (CC) of planar graphs. I thought about paths and CC.
• I saw a racoon.
• I wrote an API in Python to help us visualise algorithms for tetrises.
• We met our mentor Periklis. He has a strong personality and he is really straightforward. He suggested looking at tetrises as functions. I definitely have to try this aproach.

### Second Week

June 8th - June 14th
• I found a construction of big tetrises using mutually orthogonal latin squares (MOLS). There is some correspondence between MOLS and tetrises I would like to explore further.
• We went to New York. We saw Central park and went to the top of Rockefeller center. Very nice view.
• I am excited about MOLS. I tried to define ring operations on them. We will see what happens.
• The racoon comes every day at 9pm to eat garbage. I saw a fox too.

### Third Week

June 15th - June 21th
• I generalized the result about dense hypergraphs. An application of the generalization led to a new condition. If $\delta\Delta \geq 2(n-2)$, then the chromatic number is at most $n$.
• Other REU students taught me how to play a game called Ragecage. It was fun.
• We went to Washington DC. We saw the White House, monuments, memorials and we touched a Moon rock. I went to the National Archive and I saw the Declaration of Independence. It is really faded.

### Fourth Week

June 22th - June 28th
• I read more papers on EFL trying to find a result to improve.

### Fifth Week

June 29th - July 5th
• We rent a car.
• We went to Island Beach state Park and Atlantic City.
• We had a long trip to Niagara Falls. They are beautiful.
• I am trying to prove the EFL for a special case of "star" graphs.

### Sixth Week

July 6th - July 12th
• I failed modifying our algorithm for the "star" graphs.
• Aneta and I presented our results during the final presentation.
• We said "Bye!" and departed back home.