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### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

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### A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...

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### Why have most maximal cliques of Paley graphs odd size?

I ask this question mainly by curiosity.
See here for definitions and a plot of the clique numbers of the Paley graphs for the primes $p\equiv 1 \pmod 4$ up to $10000$.
Is there an ...

**5**

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189 views

### Van der Waerden like theorem

I am trying to develop bounds for the function B(k) where B(k) is defined as the least such positive integer so that whenever the set $\{1,2,\cdots B(k)\}$ is partitioned into two parts at least one ...

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148 views

### Weak Arithmetic Progressions

I am studying a special type of a sequence on the naturals which I am calling a weak arithmetic progression.
Formally I call a k-sequence $x_1< x_2 \cdots< x_k$ a weak arithmetic progression ...

**3**

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### Constructive lower bounds for multicolor Ramsey numbers

The $k$-color Ramsey number of the complete graph $K_n$, denoted with $R_k(n)$, is defined to be the smallest integer $t$, such that in any $k$-coloring of the edges of $K_t$, there is a complete ...

**3**

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104 views

### Bound on how many 2-colored graphs have maxredclique $i$, maxblueclique $j$

Fix $i,j$. I want a bound $F(i,j)$ on the following:
$\sum_{n=1}^\infty$ (the number of 2-colorings of the edges of $K_n$ such that
the MAX RED clique is size exactly ...

**2**

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### Slightly improving bounds on two-color Ramsey numbers by globally pruning edges and counting connected vertices in instances of two-colored complete graphs

The two-color Ramsey number, $R(m, n)$, is the minimum number of vertices, $||V||$, in a complete graph necessary for there to exist a clique of order $m$ or an independent set of order $n$. In terms ...

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### Genus tradeoffs in bipartite graph

Given $G$ as bipartite graph of genus $g(G)$ with number of vertices of each color being $N$ with $A$ as $N\times N$ biadjacency matrix. Denote $\bar{G}$ to bipartite graph of genus $g(\bar{G})$ of ...

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### Mixed Tsirelson Norm

A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...

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234 views

### Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ?

Hello All,
I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it.
Any 2-coloring of the scope of equilateral triangle must ...