Since mathematical syntax can be contrived by a neural network, I suggest writing in a way that engages human intelligence. Instead of a Platonic view that a theorem is a conjectural belief with a proof, to value the path of creating innovative ideas, we rephrase conjecture as research question and regard theorems as a profound way to summarise nontrivial thought processes. The measure of whether a key idea is useful is illustrated by what can be proven thereby, and yet in proof writing, the central ideas are usually lost in the details. Thus, to appreciate my blog, the reader is invited to be open-minded about a style that may seem non-standard, but which I contend is preferable.
An edge colouring of the complete graph
Let be a colouring of the edges of the complete graph . We suppose the number of colours in the image of is at most .
Let . There are edges incident to in , and so if , then for some , there are at least edges at of colour . By iterating this observation, it is possible to a sizeable clique with a large common neighbourhood in some colour.
The case with colour imbalance
Suppose , with colours red and blue. If the average red degree is , there are at most vertices with red degree . After deleting those, if is the size of the largest blue clique, there are red edges incident to it.
Outside this blue clique, the number of vertices with at least red neighbours in it is at most , so there are still vertices joined by a red edge to at most a proportion of the blue .
There are possible red neighbourhoods, and so there is a red neighbourhood held in common by at least other vertices. Thus, we obtain a blue clique of size at least that is joined by blue edges to as many further vertices. In the literature, such a graph is called a book.
The book lemma
Let and be disjoint subsets of . Also, let be the minimum proportion of colour neighbours of a vertex in . Fix and , . Then we can select , joined to in colour , and such that each decreases by at most , and for some and , increases by . Here, and . This is useful, because if is large, this yields a so-called density boost, and if is small, is joined to in colour and to vertices of in the same colour.
Proving this fact, which I call the book lemma, involves elementary analysis and probability. Equipped with the book lemma, one can deduce that if , where , then there is a complete subgraph of size in some colour.