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Math-842-A2 · revision r1 · on Math-842

gpt pro 5.2 attempt on Math-842

No claim recorded · LLM-generated · gpt pro 5.2 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. FORMAL RESTATEMENT
  2. QUICK LITERATURE/CONTEXT CHECK
  3. ATTACK PLAN
  4. WORK
  5. FAST REALITY CHECK (exact computation for small parameters).
  6. VERIFICATION
  7. FINAL

Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$ (for $\emptyset\neq S\subseteq A$) are monochromatic. Estimate $F(k)$. The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem . Erdős and Spencer [ErSp89] proved that\[F(k) \geq 2^{ck^2/\log k}\]for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner [BENTW17] have improved this to\[F(k) \geq 2^{2^{k-1}/k}.\] References [BENTW17] Balogh, József and Eberhard, Sean and Narayanan, Bhargav and Treglown, Andrew and Wagner, Adam Zsolt, An improved lower bound for Folkman's theorem . Bull. Lond. Math. Soc. $2017$, 745-747. [ErSp89] Erdős, Paul and Spencer, Joel, Monochromatic sumsets . J. Combin. Theory Ser. A $1989$, 162-163.

FORMAL RESTATEMENT

Fix $k\ge 1$. For $N\ge 1$, a 2-colouring of $[N]:=\{1,2,\dots,N\}$ is a map $\chi:[N]\to\{0,1\}$. For a finite set $A\subseteq [N]$, define its nonempty subset-sum set \[ \mathrm{FS}(A):=\Big\{\sum_{a\in S}a:\ \emptyset\ne S\subseteq A\Big\}\subseteq \mathbb N. \] We say $A$ is monochromatic-sum for $\chi$ if
  1. $\mathrm{FS}(A)\subseteq [N]$ (so every sum is assigned a colour), and
  2. $\chi$ is constant on $\mathrm{FS}(A)$.
Define $F(k)$ to be the minimal $N$ (if it exists) such that every 2-colouring of $[N]$ contains some $A\subseteq [N]$ with $|A|=k$ that is monochromatic-sum.

Ambiguity in the literal statement. The original phrasing does not explicitly require $\mathrm{FS}(A)\subseteq [N]$, but since colours are only defined on $[N]$, the natural convention is to require it; otherwise the property is ill-posed. I adopt that convention.

QUICK LITERATURE/CONTEXT CHECK

The statement records large lower bounds for $F(k)$ from [ErSp89,BENTW17]. Per the integrity rule I do not add or use any additional external results.

ATTACK PLAN

Proof-track ideas. (i) Compute exact values for small $k$ by exhaustive search over all colourings (reality check). (ii) Prove structural reductions (monotonicity; necessary constraints on $A$ such as $\sum A\le N$).

Disproof-track ideas. For any proposed explicit upper bound on $F(k)$, attempt to construct a colouring of $[N]$ avoiding monochromatic-sum sets of size $k$ using backtracking/SAT.

WORK

Lemma (Necessary sum constraint): If $A\subseteq [N]$ satisfies $\mathrm{FS}(A)\subseteq [N]$, then $\sum_{a\in A} a \le N$.
Proof: The full sum $\sum_{a\in A}a$ is an element of $\mathrm{FS}(A)$ (take $S=A$). If $\mathrm{FS}(A)\subseteq [N]$, then in particular $\sum_{a\in A}a\le N$. ∎
Proposition (Exact value for $k=2$): Under the convention above (distinct elements in $A$), one has $F(2)=9$.
Proof: For $k=2$, a set $A=\{x,y\}$ with $x<y$ is monochromatic-sum iff $x,y,x+y\in [N]$ and $\chi(x)=\chi(y)=\chi(x+y)$. By Lemma Lemma (Necessary sum constraint), necessarily $x+y\le N$.

Lower bound $F(2)>8$. Consider the colouring of $[8]$ given (in order $1,2,\dots,8$) by \[ BBRBRRRB, \] where $B$ means colour $0$ and $R$ means colour $1$. I checked all pairs $1\le x<y\le 8$ with $x+y\le 8$ (there are finitely many) and verified that none satisfy $\chi(x)=\chi(y)=\chi(x+y)$ under this colouring. Hence there is a 2-colouring of $[8]$ with no monochromatic-sum $2$-set, so $F(2)\ge 9$.

Upper bound $F(2)\le 9$. I exhaustively checked all $2^9=512$ colourings of $[9]$; in every case there exists a pair $1\le x<y\le 9$ with $x+y\le 9$ and $\chi(x)=\chi(y)=\chi(x+y)$. Therefore every colouring of $[9]$ contains a monochromatic-sum set of size $2$, so $F(2)\le 9$.

Combining yields $F(2)=9$. ∎

FAST REALITY CHECK (exact computation for small parameters).
The bounds in Proposition prop:531-F2 were verified by exhaustive search over all colourings of $[N]$ for $N\le 9$ and all candidate pairs $\{x,y\}$ with $x+y\le N$.

For $k=3$, exhaustive search over all colourings becomes infeasible for large $N$ because there are $2^N$ colourings, but I did check the existence of one obstruction colouring for each $N\le 25$; in particular, \[ F(3)>25. \] One explicit witness colouring of $[25]$ avoiding monochromatic-sum $3$-sets is (in order $1,2,\dots,25$) \[ RRBBRBRRRRBBBBBBBBBBBBBBB. \] This was verified by enumerating all triples $A=\{a<b<c\}\subseteq [25]$ with $a+b+c\le 25$ (Lemma Lemma (Necessary sum constraint)) and checking that the seven subset sums in $\mathrm{FS}(A)$ are not all the same colour.

VERIFICATION

  • Definition check: $\mathrm{FS}(A)$ uses distinct elements, so $k=2$ forbids using $x=x$; this is why $F(2)$ differs from the classical "monochromatic $x+y=z$" condition that allows $x=y$.
  • Proposition prop:531-F2: both the lower-bound witness on $[8]$ and the universal property on $[9]$ were established by exhaustive enumeration over finite search spaces.
  • For $F(3)>25$: the verification checks all eligible triples for the displayed colouring, so it is exact for that $N$.

FINAL

UNRESOLVED

(i) Strongest proved partial result. Exact small-$k$ information: $F(2)=9$ (Proposition prop:531-F2) and an explicit computed lower bound $F(3)>25$ with a witness colouring. Additionally, any candidate $A$ must satisfy $\sum A\le N$ (Lemma Lemma (Necessary sum constraint)).

(ii) First gap (crisp statement). Determine $F(3)$ (even within reasonable magnitude), or prove any explicit general upper bound on $F(k)$ as a function of $k$.

(iii) Top 3 next moves. 1. Implement SAT/ILP with symmetry-breaking for $k=3$ to push explicit lower-bound colourings to much larger $N$, narrowing the range for $F(3)$. 2. Seek a finitary proof of existence of $F(k)$ with explicit quantitative bounds by encoding the problem as a finite system of partition-regular equations and tracking constants. 3. Attempt an iterative "finite sums" construction that produces a monochromatic $\mathrm{FS}(A)$ set while explicitly controlling $\max \mathrm{FS}(A)\le N$, yielding constructive upper bounds.

(iv) Minimal counterexample structure. A colouring witnessing large $F(k)$ must avoid monochromaticity on every $\mathrm{FS}(A)$ with $|A|=k$, forcing a dense pattern of colour alternation across many additive configurations; for $k=3$ this means avoiding monochromaticity on all $7$-term sets $\mathrm{FS}(\{a,b,c\})$ with $a+b+c\le N$.