I learned more about chernoff bounds and markov/chebyshev inequalities today, I was getting tired of glossing my eyes over all these probabilistic arguments that seem way too powerful. I was surprised that the proof of the chernoff bound was actually a very simple calculation and a pretty intuitive result. Anyways, that's another tool in the toolbox I hope to use, part of the reason I wanted to familiarize myself with these was that I think it may be helpful in understanding this conflict graph, which is really turning out to be the first main problem in this project. So I have two routes I am considering, using a probabilistic/heuristic method to get some sort of bound on the theta function, which is pretty appealing since earlier I was thinking that when I started this project, I really wanted some way to connect the expansion properties of the 3XOR instance to the analysis of the local hamiltonian. Now I realize that maybe the right way to do that is indirectly, expansion for Grigoriev's 3XOR instance is just a result of the fact that the graph was sampled randomly with the right parameters, so if I treat the conflict graph as that, perhaps some specific version of an Erdos-Renyi random graph, I would be able to use the known results about the theta function for random graphs and I would be making use of the same properties that underly the graphs expansion. One caveat though, the clause-variable graph is the one sampled from the erdos-renyi distribution, the conflict graph is induced from it but its certainly not pulled from the same distribution. Although I did some calculations today on the expected degree of the conflict graph and it doesn't depend on n, so its like approximately d-regular for some constant d, giving me a little bit of hope that it really is just Erdos-Renyi with some wierd parameters. Anyways thats route 1, and another thing that makes this route appealing is that the tools to bound the theta function for random graphs uses ideas from random matrix theory, which I have been wanting to learn about. Route 2, is what seems like the hard way, I would have to get more than a surface level understanding of these Johnson association schemes. In particular, I need to understand how exactly people have gotten those beautiful combinatorial formulas bounding the theta number for these specific johnson graphs, and then determine whether or not I can generalize those ideas to my weird bipartite mess, or if my weird bipartite mess already fits into that framework which is not at all obvious right now. Today was good overall! I spent most of the time writing so that I could submit the poster idea so it feels like I didn't get a lot of thinking done but I actually feel a lot more clear headed about the whole project now. On top of that I actually got a mini result as well, I did by hand the probabilistic calculation for the approximate degree of the conflict graph, and I got approximately 450 when m = 100n! Pretty cool number I guess, what I am starting to think now is like how I can characterize these conflict graphs from the clause variable structure alone, that would streamline this process if I want to try other hamiltonians and then I could say something like, given this type of 3xor instance, its local hamiltonian has this bound on the eigenvalue which comes from a simple SDP (Lovasz Theta). I just love saying "The Lovasz Theta Function".