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Statement

Setting (p. 163). $P(\cdot)$ is the set of sums of distinct elements, as on p. 162. In the $(r,s)$ sumset game Player 1 picks distinct $a_1,\ldots,a_r\in\mathbb{N}$; Player 2, seeing them, then picks $a_{r+1},\ldots,a_{r+s}\in\mathbb{N}$, distinct from each other and from the earlier $a_i$; the payoff to Player 1 is $|P(a_1,\ldots,a_{r+s})|$. $V(r,s)$ is the value of this perfect-information game.

Conjecture (p. 163, unnumbered, quoted). "Can an exact formula for $V(r,s)$ be found? We conjecture $V(r,s)\ge cs^22^r$. Note $V(r,s)\le\binom{s+2}2 2^{r-1}$ as Player 2 may select $2a_1,\ldots,(s+1)a_1$."

The constant $c$ is not specified further on p. 163. The paper adds, as a heuristic: "Perhaps Player 1 can pick $r$ numbers sufficiently independent so that Player 2 can do no better."

Source. P. Erdős and J. Spencer, Monochromatic sumsets, J. Combin. Theory Ser. A 50 (1989), 162--163: printed p. 163. The edition read is identified on the source card.

Read depth. Claims checked: the game's definition, the conjecture and the upper-bound remark were read clause by clause on the printed page. Nothing here is independently reviewed.

Proof pointer

The conjecture is open in the paper. The upper bound is the paper's one-line remark: Player 2 answers with the multiples $2a_1,\ldots,(s+1)a_1$.

Dependencies

None.

Bears on

• Problem 531: the paper says the game arose from attempts to remove the $\lg k$ factor from the exponent of its lower bound for the Folkman function (theorem page); it does not state what bound the conjecture would give.