Sam Nead and David White have given very good answers so far. I want to expand on some related aspects. I've done a research in graph theory with high school students before which lead to a just published paper so I have some experience here. (I've done more work of a similar sort with number theory and high school students.)

One thing one needs to be careful about, is that almost any problem that is listed in an open problems collection has had a fair number of people look at it before. So one thing that is important to do is to try to answer not the whole question, but look at some manageable part.

A related issue is that to be blunt, some people who do research mentorship with high school and undergraduate students hoard problems since finding really good problems for this can be tough. I've shared things with students who aren't mine before, but that's generally been when I've had a glut of problems, which is rare. That said, I do have a specific research problem which I'm going to just throw out here because it is both somewhat on the topic, and because I have a bunch of others currently in my stash, and I'm not going to have another graph theory student group for a while. This assumes some small familiarity with the Hadwiger-Nelson problem.

I call this project "How not to solve the Hadwiger-Nelson problem."

Coloring a hex grid carefully gives $HN(2) \leq 7$, that is the chromatic number of the unit graph on the plane is at most 7. One can try the same thing with other grids, but for the other two obvious ones, triangle and square grids one gets a worse result. To some extent this seems to be due to the hex grid being the most well behaved grid in many respects.

What happens if we insist on coloring tilings of the plane and use even more irregular shapes? Regular pentagons cannot tile the plane, but there are 15 families of irregular pentagons which do tile the plane. One project direction is to see what numbers they give (almost certainly pretty bad) for the Hadwiger-Nelson bound.

There are other tiles which may be of interest here, such as the standard octagon with square grid pattern.

Recently, David Smith discovered a set of so-called Einstein tiles. An Einstein (German for one-stone) was a long sought for single tile that could completely tile the plane and do so aperiodically. These tiles are very irregular and so one would expect that they give even worse bounds. How much worse are they?

Sam Nead and David White have given very good answers so far. I want to expand on some related aspects. I've done a research in graph theory with high school students before which led to a just published paper so I have some experience here. (I've done more work of a similar sort with number theory and high school students.)

One thing one needs to be careful about, is that almost any problem that is listed in an open problems collection has had a fair number of people look at it before. So one thing that is important to do is to not try to answer the whole question, but look at some manageable part.

A related issue is that to be blunt, some people who do research mentorship with high school and undergraduate students hoard problems since finding really good problems for this can be tough. I've shared things with students who aren't mine before, but that's generally been when I've had a glut of problems, which is rare. That said, I do have a specific research problem which I'm going to just throw out here because it is both somewhat on the topic, and because I have a bunch of others currently in my stash, and I'm not going to have another graph theory student group for a while. This assumes some small familiarity with the Hadwiger–Nelson problem.

I call this project "How not to solve the Hadwiger–Nelson problem."

Coloring a hex grid carefully gives $\operatorname{HN}(2) \leq 7$, that is the chromatic number of the unit graph on the plane is at most 7. One can try the same thing with other grids, but for the other two obvious ones, triangle and square grids one gets a worse result. To some extent this seems to be due to the hex grid being the most well behaved grid in many respects.

What happens if we insist on coloring tilings of the plane and use even more irregular shapes? Regular pentagons cannot tile the plane, but there are 15 families of irregular pentagons which do tile the plane. One project direction is to see what numbers they give (almost certainly pretty bad) for the Hadwiger–Nelson bound.

There are other tiles which may be of interest here, such as the standard octagon with square grid pattern.

Recently, David Smith discovered a set of so-called Einstein tiles. An Einstein (German for one-stone) was a long sought for single tile that could completely tile the plane and do so aperiodically. These tiles are very irregular and so one would expect that they give even worse bounds. How much worse are they?