I've just started studying combinatorics and I'm having a bit of trouble understanding the definition of Ramsey's Theorem that my book gives me. It states,
Let $q_1,q_2,...,q_n,t$ be positive integers with $q_1 \geq t, q_2 \geq t,... q_n \geq t.$ Then there exists a positive integer, and thus a smallest positive integer $N(q_1,q_2,...q_n; t)$, depending only on $q_1, q_2,...q_n,$ and $t$, with the following property. If $m \geq N(q_1, q_2,...,q_n;t)$ and $S$ is a set of $m$ objects such that the $t$-element subsets of $S$ are distributed among $n$ boxes, then either there exist $q_1$ objects such that all $t$-element subsets of these $q_1$ objects are in the first box, or there exist $q_2$ objects such that all $t$-element subsets of these $q_2$ objects are in the second box, ..., or there exist $q_n$ objects such that all $t$-element subsets of these $q_n$ objects are in the $n$th box
My main question is with the $N(q_1,q_2,...q_n; t)$ notation. Is this similar to a function where $N(q_n)$ denotes a relation with $q_n$ as the input? Further, why is there a semicolon separating the $t$? What is this supposed to represent?
Also, how is the $t-$element subset distributed? If I understand correctly, a $t-$element subset is a subset of $S$ with $t$ elements. Does this mean that each element of the $t-$element subset is put into a box (analogous to the pigeonhole principle where $n+1$ elements are put into $n$ boxes)?
Yes, $N$ is a function here. If we want to do it rigorously, then the domain of $N$ is the following subset of $\Bbb N^*\times \Bbb N$, where $\Bbb N^*=\bigcup_{i=1}^{\infty}\Bbb N^i$ is the set of all finite sequences of natural numbers. $$D=\{((q_1,\dots,q_n),\, t) \, :\, 0 <t\le\min_iq_i\}$$ Then $N(q_1,\dots,q_n; t)$ is shorthand for $N\big(((q_1,\dots,q_n), t)\big)$.
No, the boxes contain the $t$-element subsets of $S$, not the elements of these, so that we have $\binom mt$ objects (the $t$-element subsets) for $n$ boxes.