we-solve (beta)

Math-5771-A1 · revision r1 · on Math-5771

Reused writeup on Math-5771 (gpt pro 5.2)

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. Erdős Problem 617 --- collection record
  2. PROVENANCE AND STATUS
  3. REUSED SOURCE 1
  4. COMPLETION ESTIMATE

Erdős Problem 617 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/617

Saved collaborative-database status: falsifiable. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/617.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/617.tex.

FORMAL RESTATEMENT Fix an integer $r\ge3$. Consider an $r$-edge-coloring of the complete graph $K_{r^2+1}$. The conjecture asks whether there must exist a set $S$ of $r+1$ vertices such that the induced $K_{r+1}$ on $S$ is missing at least one of the $r$ colors.

Equivalently, there is no "balanced" $r$-coloring of $K_{r^2+1}$ in which every $K_{r+1}$ uses all $r$ colors.

QUICK LITERATURE/CONTEXT CHECK I do not use external sources; I only use the statements recorded in the problem file. The file states:

  • The statement is false for $r=2$.
  • Erdős and Gyárfás showed it becomes false for infinitely many $r$ if $K_{r^2+1}$ is replaced by $K_{r^2}$.
  • The conjecture has been proved for $r=3$ and $r=4$.

ATTACK PLAN Translate the "missing a color" condition into graph properties of each color class. Then:

  • Produce explicit balanced colorings for the variant $K_{r^2}$ (to match the known failure of that variant).
  • Derive necessary constraints for any putative balanced coloring of $K_{r^2+1}$.
Also check small cases computationally where feasible.

WORK Lemma 1 (balanced $\Leftrightarrow$ small independence number in each color class). Let the edges of $K_n$ be colored with colors $\{1,\dots,r\}$. For a fixed color $c$, let $H_c$ be the spanning subgraph whose edges are exactly those of color $c$. Then the coloring is balanced (i.e., every $(r+1)$-set of vertices contains at least one edge of each color) if and only if \[ \alpha(H_c)\le r\quad\text{for every }c\in\{1,\dots,r\}. \]

Proof. Fix a color $c$. A set $S$ of $r+1$ vertices has no edges of color $c$ if and only if $S$ is an independent set in the graph $H_c$. Therefore the condition "every $(r+1)$-set has at least one $c$-colored edge" is exactly the condition "$H_c$ has no independent set of size $r+1$", i.e. $\alpha(H_c)\le r$. Imposing this for every color $c$ is equivalent to balancedness. $\square$

Lemma 2 (explicit balanced $3$-coloring of $K_9$ for the $r^2$ variant). There exists a $3$-coloring of the edges of $K_9$ in which every $K_4$ contains all $3$ colors. In particular, the conjectured statement becomes false if one replaces $K_{r^2+1}$ by $K_{r^2}$, already for $r=3$.

Proof (explicit construction + exhaustive verification). Label the vertices by $\{0,1,2,3,4,5,6,7,8\}$. Color the $36$ edges of $K_9$ with colors $0,1,2$ as follows.

Color $0$ (18 edges): \[ \{01,04,05,06,12,13,14,23,26,28,35,37,47,48,56,57,68,78\}. \] Color $1$ (9 edges): \[ \{02,07,15,18,27,34,36,46,58\}. \] Color $2$ (9 edges): \[ \{03,08,16,17,24,25,38,45,67\}. \] (Here $ij$ denotes the edge $\{i,j\}$.)

I verified by an exhaustive script that for every $4$-subset $S\subseteq\{0,\dots,8\}$ (there are ${9\choose 4}=126$ of them), the $6$ edges induced by $S$ contain at least one edge of each of the three colors. Therefore this is a balanced $3$-coloring of $K_9$. $\square$

VERIFICATION FAST REALITY CHECK 1 (the stated $r=2$ failure). For $r=2$, balanced means "no monochromatic triangle". Label $K_5$ vertices $0,1,2,3,4$ in cyclic order. Color the $5$ cycle edges $01,12,23,34,40$ red and the $5$ diagonals blue. Every triangle in $K_5$ contains at least one cycle edge and at least one diagonal, so it uses both colors. Hence the conjecture would indeed be false if one allowed $r=2$.

FAST REALITY CHECK 2 (the $r^2$ variant for $r=3$). The coloring in Lemma 2 was checked exhaustively over all $126$ induced $K_4$'s; no exceptions were found.

FINAL UNRESOLVED (i) Strongest proved partial results here: the structural equivalence in Lemma 1, and an explicit balanced coloring for the $r^2$-vertex variant at $r=3$ (Lemma 2), matching the problem file's statement that the $r^2$ variant fails for infinitely many $r$.

(ii) First gap (crisp): prove (or disprove) that for every $r\ge3$ there is no balanced $r$-coloring of $K_{r^2+1}$ (i.e., show that some $(r+1)$-set is missing a color in every $r$-coloring).

(iii) Top 3 next moves: 1. Use Lemma 1 to translate the conjecture into constraints on an edge-partition of $K_{r^2+1}$ into $r$ graphs $H_c$ each having $\alpha(H_c)\le r$; attempt to show such a partition is impossible at $n=r^2+1$. 2. Develop a counting argument on $(r+1)$-subsets: for each color $c$, count how many $(r+1)$-sets are independent in $H_c$ and show that the sum over colors must be positive. 3. Computational exploration for the next open values of $r$ (beyond those known in the file), searching either for a balanced coloring (as a disproof) or for a certificate that none exists.

(iv) Minimal counterexample structure: A minimal counterexample to the conjecture would be a balanced coloring of $K_{r^2+1}$. Equivalently, it would be a partition of the edges into $r$ color-class graphs $H_1,\dots,H_r$ on $n=r^2+1$ vertices with $\alpha(H_c)\le r$ for every $c$. Such an object would likely have strong algebraic symmetry (similar in flavor to finite-geometry constructions) and would in particular give an extremal example for the independence constraints of Lemma 1.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.