How do I explain Erdos-Szekeres theorem to a child? So,a $5$th grader yesterday asked me to explain the Erdos- Szekeres Theorem to him.He is not too proficient with series and so having problem to understand my explanation.How do I simply explain it to with with examples so that he may understand it?
Thanks for any suggestions..
 A: I’m assuming that you’re trying to explain what it says; explaining why it’s true is considerably harder.
You can illustrate the setting with simple graphs: a term $x_k$ in the finite sequence is plotted as a point $\langle k,x_k\rangle$. Draw several of these graphs and illustrate some of the monotonic subsequences, both increasing and decreasing. For simplicity you can take the terms to be integers. Once the idea is clear, that we’re trying to find the longest possible monotonic subsequence, you can illustrate the boundary case:
                                               *  
                                                 *  
                                                   *  
                                                     *  
                                      *  
                                        *  
                                          *  
                                            *  
                             *  
                               *  
                                 *  
                                   *  
                    *  
                      *  
                        *  
                          *  

It’s not too hard to explain that any set of five points has to hit at least two of the falling blocks and therefore cannot be decreasing, but also has to have at least two points in one block and therefore cannot be increasing. It’s also easy to explain how this generalizes to $n^2$ points for any $n$. Now you can explain that the theorem itself simply says that adding just one more point guarantees you a monotonic subsequence of length $5$, no matter how the first $16$ were placed.
You can also phrase it in terms of a line of people of different heights: if there are, say, $10$ people in the line, then we can always find $4$ whose heights are either decreasing or increasing as we go from front to back of the line.
A: I would start with the infinite version: every infinite sequence (not series) has a monotone subsequence (https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem). Do examples. Have him try to find counterexamples. Then perhaps you can get him to see that there might be a finite version that quantifies things.
I may try this next week with a group of fifth graders. Since we've spent time thinking about the pigeonhole principle we might manage a proof.
