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I'm currently a PhD student in theoretical computer science. I've been working on this problem daily for almost a month that has been well studied and was assigned to me by my advisor. The problem is specifically about establishing an upper bound on coloring a special class of graphs with some interesting properties. However, the last major improvement on the upper bound dates almost 20 years ago in SODA.

As I am sure some of you are aware, establishing upper bounds on coloring involves "playing" around with different coloring constructions and trying to establish an upper bound for that construction. However, so far, all my constructions (that I achieved without reading any literature on the problem) have lead to ever so slightly worse bounds to the existing literature (I know the best current bound).

My question is, when is a good time to give up and start reading other people's work? I have tens of ideas on different coloring arguments that I want to explore, but I am not sure if I am stuck in "naive" land using tools that will get me nowhere. Should I just call it quits and start reading what other people have done? My advisor told me he would be supportive in both decisions (which I am very thankful for), but that ultimately the decision is mine (although he prefers me starting to read). Any advice?

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    $\begingroup$ I'm not sure why reading other people's work would be giving up? My advice would be to almost always spend some proportion of your time reading other people's work (or at least scavenging through them for useful results and proof techniques). The question is then how much time per day/week/month you want to allocate to that versus trying to tackle the problem at hand. Could be as little or as much as suits you, and doesn't have to be a constant. (I'm also a PhD student, perhaps people with more experience will disagree with me) $\endgroup$
    – Tassle
    Nov 29, 2021 at 7:41
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    $\begingroup$ IMO, the only possible downside to reading existing work is if it "boxes" you in some range of ideas, and limits your imagination in new constructions. But this can be avoided, if you write down your existing ideas, and possibly devote some time for exploring new stuff. However, this small downside is considerably outweighed by the benefit of reading existing works, and learning techniques and tools that would augment your mathematical toolbox. So in short - I think reading is the best course of action. $\endgroup$
    – Shaull
    Nov 29, 2021 at 7:42
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    $\begingroup$ When you get to the point of trying to get your results published, you will have to know the existing literature so that you can give proper attribution and references to what was already known, and to be able to explain what is new and original in your work. Doing it now rather than afterwards will save you from spending lots of effort on reinventing the wheel. $\endgroup$ Nov 29, 2021 at 13:59

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Scenario 1: You spend several months tinkering around with colorings, not reading any literature. After many failed attempts you finally discover one that works. Before you can write a paper about it, you need to study literature so that you can properly situate your result. You feel bitter and lonely for some reason.

Scenario 2: You read the literature, but not too seriously, and you tinker with colorings as well. By seeing what other people have done, you eliminate a couple of possible approaches, which saves you a month or two. One paper leads you in the right direction. You find your solution faster. And when it's time to write up the paper, you already know the literature. The author of the helpful paper contacts you, because they're delighted their work was useful to you. They invite you to talk at a seminar, where you meet your future spouse.

Scenario 3: You decide to read all literature first. You die of old age extremely wise.

What's the question again?

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My advice would be to follow the preference of your advisor and start reading the literature. Your advisor knows you and he selected the problem for you. That he lets the decision to you but expresses a preference, this is a good sign to me, regarding your advisor/advisee relationship.

Disclaimer: I am just a random person from the internet, don't blindly follow my advice 🙃

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