I am currently studying the theory behind Well-Quasi-Orders. However I am having some issues in understanding how an infinite anti-chain can be produced to disprove the claim that a partial order $P$ is a w.q.o.
In particular I am wondering whether it is logically sound to present as proof not the infinite anti-chain but rather an algorithm to produce it.
More specifically, I've been trying to solve an exercise from the following lecture which asks whether the class of $P_{3}\text{-free}$ graphs is a w.q.o or not on the induced subgraph operation $\leq_{i}$
I've proved that said class contains unions of cliques and defined an ordering of this set as follows $P=\lbrace G_{0},G_{1},G_{2},.... \rbrace$ with $G_{i}$ representing all $P_{3}\text{-free}$ graphs on $i$ vertices (and graphs in $G_{i}$ being represented in a touple-like alpharithmetic form $(a_{1},....,a_{i})$ such that $a_{1} \leq a_{2} \leq.... \leq a_{i}$ to avoid repetitions and $\sum_{j=1}^{i} a_{j}=i$ to define an order. For example $(0,0,2,2)$ $\in G_{4}$ encodes the graph with two disconnected $P_{2}$ components.
I have defined an algorithm $B$ that produces an anti-chain that grows bigger and bigger at each step
The algorithm uses gadgets defined as $B_{a,c}$ which represent the graph with $c+1$ connected components, the first $c$ of which are $K_{1}$(i.e a sole vertice) and the last component representing $K_{a}$
The steps of my algorithm are the following:
Define $L$ as the sequence
Add $B_{a,0}$ to $L$ ($a$ any number $>$ 0)
repeat $\infty$ times:
4.Replace each item $B_{a,c} \in L$ by $B_{a^{'}=a+1 + 10^{6},c}$ and produce $L^{'}$ this way.
5.Let $X=B_{a^{'},c}$ be the graph with the maximum $c$ amongst those in $L^{'}$
6.Add to $L=L^{'}$ the graph $B_{a^{'}-2,c+1}$
Certainly, this algorithm does create an anti-chain( in each step I create a graph with more connected components yet less vertices). However I am not sure that such a technique is valid for disproving w.q.o on the infinite anti-thesis claim because of the fact that it works on a step by step basis.
Any help is appreciated. Thank you very much!