**Definition.** $h(n\_1,n\_2)$ is the least number $m$ such that, if the edges of $K\_m$ are colored with two colors, $1$ and $2,$ then for some color $i\in\{1,2\}$ there is a set $W\subseteq V(K\_m)$ such that $|W|=n\_i$ and every triangle in $W$ has an odd number of edges of color $i;$ **in other words,** for some $i\in\{1,2\},$ the graph consisting of the edges of color $i$ has an induced subgraph $H$ of order $n\_i$ such that $H$ has at most two components, and each component of $H$ is a clique.



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> **Question 1.** Is there any literature on $h(n\_1,n\_2)$ ?  
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> **Question 2.** Is $h(4,5)=8$ ?
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Here are some easy bounds for $h(n\_1,n\_2)$ in terms of ordinary Ramsey numbers.


**Definition.** The Ramsey number $R(n\_1,n\_2;d)$ is the least number $m$ such that, given an $m$-element set $V$ and any set $S\subseteq\binom Vd,$ we can find a set $W\subseteq V$ such that either $|W|=n\_1$ and $\binom Wd\cap S=\emptyset,$ or else $|W|=n\_2$ and $\binom Wd\subseteq S.$


**Definition.** As in my previous question [A funny kind of Ramsey number](https://mathoverflow.net/questions/300018/a-funny-kind-of-ramsey-number), $f(n)$ is the least number $m$ such that, given an $m$-element set $V$ and any set $S\subseteq\binom V3,$ we can find an $n$-element set $X\subseteq V$ such that, for each $4$-element set $Y\subseteq X,$ we have $|\binom Y3\cap S|\equiv0\pmod2.$


**Upper bound:** $h(m+1,n+1)\le R(m,n;2)+1.$


**Lower bound:** $f(h(m,n))\ge R(m,n;3).$


Regarding $h(4,5),$ I know that $$8\le h(4,5)\le10.$$
On the one hand, $h(4,5)\le R(3,4;2)+1=10.$ On the other hand, to see that $h(4,5)\gt7,$ take a Hamiltonian cycle $C$ in $K\_7$ and color the edges of $C$ with color $2$ and the rest with color $1.$ (This is the simplest of a whole bunch of $7$-point counterexamples.) I have not found a counterexample to the conjecture that $h(4,5)=8.$



Another funny kind of Ramsey number
The statement $h(4,5)=8$ is true. It can be checked by enumerating all 8-vertex graphs (a total of 12346) and check them one by one. 


For those who want to double-check, there are 48 graphs in which there is exactly one instance of $W$, and 43 in which there are two.


