Canonical versions of van der Waerden and Szemerédi type recurrence (conjectures and implications)

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Problem: Reconstructed statement: for every $\ell\in\mathbb N$ and every countable partition $\mathbb N=\bigcup_{i=1}^\infty C_i$, there is $d\in\mathbb N$ such that either

1. some cell $C_{i_0}$ contains an arithmetic progression

or

2. every arithmetic progression $a,a+d,\dots,a+\ell d$ is rainbow, equivalently for every $i$ and every $0\le j<k\le \ell$,

Here $A-t=\{n:n+t\in A\}$. This is exactly the statement in the supplied conjecture.

Result: The conjecture is false for every $\ell\ge 2$.

Construct a countable partition of $\mathbb N$ as follows. For each $d\ge1$, set

These pairs are pairwise disjoint: if $d<e$, then

so $P_d$ lies entirely before the first element of $P_e$. Now let the partition consist of all pairs $P_d$, together with singleton sets for every remaining natural number.

Every cell has size at most $2$. Hence for any $\ell\ge2$, no cell contains an arithmetic progression of length $\ell+1\ge3$. Thus alternative (i) fails for every $d$.

On the other hand, for each $d\ge1$, the cell $P_d$ contains two elements differing by exactly $d$:

Therefore

so alternative (ii) fails for this same $d$, already with $j=0,k=1$.

Thus for every $d\in\mathbb N$, neither (i) nor (ii) holds. This disproves the conjecture.

The failure is caused by allowing arbitrary infinite partitions with very small cells. A natural repair would need some largeness assumption on the cells, or else restrict to finite partitions; under finite partitions, ordinary van der Waerden already gives alternative (i).

Citation: No external citation is needed; the counterexample above is elementary.