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The answer is NO, at least for $n=4$. I show here that $\eta(4) \leq 7$. So let $\omega$ be a valuation on $K_7$ ; we will show that it is $4$-valid. First, we need some notation : for $A \cup B$ a …

Let $d$ be an integer $\geq 2$, and let $\Omega = \lbrace 0,1 \rbrace^d$, $A \subseteq \lbrace 0,1 \rbrace^2 $ and $i,j$ integers with $1 \leq i < j \leq d$. If we select an element $(x_1,x_2, \ldots …

By a well-known probabilistic argument due to Erdos, if $k>1$ is an integer then for all large enough $n$, there an asymmetric relation $R$ on $X=\lbrace 1,2, \ldots ,n \rbrace$ (i.e. $R \subseteq X^2 …

Let $\cal F$ denote the group of all finitely-supported permutations of $\mathbb N$. Say that a finite subgroup $G$ of $\cal F$ is singular if $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ but n …

If I understood the OP correctly, the problem can be stated as follows : Problem 1. Let $X$ be a set, let $F:{\cal P}(X) \to X$, and let $A$ be defined as above: $$A=\lbrace F(Z) | Z\subseteq X, F(Z) …

Definition I say that a function $f : \lbrace 1, \ldots ,n \rbrace \to \frac{\mathbb Z} {n \mathbb Z} $ is multiplicative iff whenever $x=yz$ for $x,y,z$ in the range of $f$, then $f(x)=f(y)+f(z)$ i …

Roth's theorem provides an estimate for the largest size of a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ (a set of integers is nonaveraging if it does not contain any nontrivial three-te …

The question on games and mathematics that appeared recently on mathoverflow (Which popular games are the most mathematical?) reminded me of a problem I encountered some time ago : starting with the i …

A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal cardinali …

I don't know if you're still interested in this problem Hailong, but here is a partial result. I make two natural restrictions : Restriction 1 : avoid small primes. Let $B_n$ denote the (infinite) …

A The Sudoku game admits a broad generalization as follows : let $r$ be an integer $\geq 2$ and let $X$ be a finite set, and ${\cal X}$ be a collection of $r$-subsets of $X$ (i.e, a $r$-uniform hype …

A special case says it all ... Let $ w_1 < w_2 < \ldots < w_{12} $ be an increasing sequence of $12$ integers ("weights") such that the total weight $W=\sum_{k=1}^{12}w_k$ is even. Say that $I \subs …