Effectiveness of Hindman’s Theorem for Bounded Sums

Abstract

Abstract. We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let HT ≤n k denote the assertion that for each k-coloring c of N there is an infinite set X ⊆ N such that all sums x∈F x for F ⊆ X and 0 < |F | ≤ n have the same color. We prove that there is a computable 2-coloring c of N such that there is no infinite computable set X such that all nonempty sums of at most 2 elements of X have the same color. It follows that HT ≤2 2 is not provable in RCA 0 and in fact we show that it implies SRT 2 2 in RCA 0 . We also show that there is a computable instance of HT ≤3 3 with all solutions computing 0 ′ . The proof of this result shows that HT ≤3 3 implies ACA 0 in RCA 0 .

New bounds on the strength of some restrictions of Hindman’s Theorem

“…In this section we first show that HT ≤2 implies ACA 0 (hence RT 3 2 ) over RCA 0 . This improves on the main result of [15] that HT ≤3 implies ACA 0 . In particular we show that HT ≤2 4 implies ACA 0 .…”

“Weak yet strong” restrictions of Hindman’s Finite Sums Theorem

“…Recently there has been some interest in the computability-theoretic and proof-theoretic strength of restrictions of Hindman's Theorem (see [10,5,4]). While [10] deals with a restriction on the sequence of finite sets in the Finite Unions formulation of Hindman's Theorem, both [5] and [4] deal with restrictions on the types of sums that are guaranteed to be colored the same color.…”

“…Let us denote by HT ≤n r the restriction of the Finite Sums Theorem to colorings with r colors and sums of at most n terms. The conjecture discussed in [1] is then that the complexity of HT ≤n r is growing with n. The main result in [5] is that the above described ∅ ′ lower bound known to hold for the full Hindman's Theorem already applies to its restriction to 4 colors and to sums of at most 3 terms (HT ≤3 4 in the notation introduced above). On the other hand, note that no upper bound other than the upper bound for full Hindman's Theorem is known to hold for this restricted version, and the same is true for HT ≤2 2 , the restriction to sums of at most 2 terms!…”

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

“…We first show that it is very easy to establish an upper bound on AHT 2 and AHT. This should be contrasted with the case of Hindman's Theorem restricted to sums of at most two terms (HT ≤2 2 in the notation of [7]), for which no upper bound other than ACA + 0 is currently known.…”

“…Recently there has been some interest in the strength of restrictions of Hindman's Theorem (see [9,7]). Interestingly, Dzhafarov, Jockusch, Solomon and Westrick [7] proved that the only known lower bound on Hindman's Theorem already hits for HT ≤3 4 (Hindman's Theorem restricted to 4-colorings and sums of at most 3 terms) and that HT ≤2 2 (Hindman's Theorem restricted to 2-colorings and sums of at most 2 terms) is unprovable in RCA 0 ). However, no upper bounds other than those known for the full Hindman's Theorem are known for HT ≤2 2 , let alone HT ≤3 4 .…”