Arithmetic progressions of polygonal numbers

Recently I have thought a lot about the square numbers and in particular arithmetic progressions of squares. Today I wondered what properties of squares followed over to triangle numbers. Fermat proved that there exists no 4 squares in arithmetic progressions. My first question was whether or not there exists an arithmetic progression of 4 triangle numbers. It turns out there is not and the proof is very simple.

Suppose that $t$ is a triangle number. Then there exists $n\in\mathbb{N}$ such that $t=\frac{n(n+1)}{2}.$ Then $8t+1=4n^{2}+4n+1=(2n+1)^2.$ Therefore if $t_{1},t_{2},t_{3},t_{4}$ is an arithmetic progression with common difference $d$ then $8t_{1}+1,8t_{2}+1,8t_{3}+1,8t_{4}+1$ is an arithmetic progression of squares with common difference $8d$ which is a contradiction.

Triangle and square numbers are both types of polygonal number (the 3-gonal and 4-gonal numbers respectively). Let $n\geq3.$ Then one can show that there exists no arithmetic progression of 4 n-gonal numbers.

My next thought was to what happened if I looked at the union of square and triangular numbers. Could I get an arithmetic progression of length 4? How about arbitrarily long? I have yet to answer the first question but I expect one exists (probably trivially). I have however answered the second question by showing that there exists no arithmetic progression of squares and triangles of length 35. The proof uses Van de Waerdens theorem:

A consequence of van de Waerdens theorem is that if you colour the integers from 1 to 35 in two different colours then there must exist a monochromatic arithmetic progression of length 4. Suppose that there exists an arithmetic progression of length 35 of triangles and squares. Then colour the triangle numbers red and the squares blue. Lift this to a colouring of integers from 1 to 35. Then there exists a monochromatic arithmetic progression of length 4. Lift this back to our squares and triangles. Then there exists an arithmetic progression of squares OR triangles of length 4 which is a contradiction.

You can generalise this the union of 2 sets of polygonal numbers. The biggest arithmetic progression I have found so far is of length 5. If you take the union of the squares and the 409-gonnal numbers then 49,169,289,409,529 is such an arithmetic progression.

My question is whether 35 is the best we can get for an upper bound. I suspect it can be drastically decreased. Mainly because I have researched embedding squares into long arithmetic progression and they usually have to take up a small density. As polygonal number or just aphine transforms  of the squares they should also. But if we reach 34 say then one of them must have density one half which doesn't feel right to me.

As a closing remark if we take the union of $n$ sets of polygonal numbers then they will not contain arbitrarily long arithmetic progressions. In fact it is bounded by the Van de Waerden number W(n,4). More generally let $\{T_{i}\}$ be a finite collection of sets which do not contain arbitrarily long arithmetic progressions. Then $T=\bigcup T_{i}$ does not contain arbitrarily long arithmetic progressions.