Research task (STEM):
Find a series of olympiad math problems, which build progressively in difficulty and lead up towards Ramsey's theorem.  At each step, add intuition for how one might reach the solution.

Evaluation rubric:
criterion: The response presents at least 3 well-formed Olympiad-level problems (e.g., (1) Let n and T be positive integers. James has 4n marbles with weights 1, 2, . . . , 4n.
He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most T. Find, in terms of n, the minimum positive integer T such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles (IMO 2024), (2) Let N be a positive integer and let a1, a2, . . . be an infinite sequence of positive integers. Suppose that, for each n > N , a_n , an is equal to the number of times a_{n-1} appears in the list a1, a2, . . . , a_{n-1}. Prove that at least one of the sequences a1, a3, a5, . . . and a2, a4, a6, . . . is eventually periodic (IMO 2024), (3) A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles triangles with two odd sides.(IMO 2006), (4) Prove that, for every positive integer n, there exists an integer m such that 2m + m is divisible by n.).
weight: 5
axis: Explicit Criteria

criterion: The response orders the problems in a way that increases in conceptual difficulty (i.e., Simple Pigeonhole Principle Problem (e.g. N number of holes and we want to fit N+1 pigeons into the holes) -> Advanced Pigeonhole Principle Problem (e.g., In any group of n≥2 people, there are at least two people who have the same number of friends) -> Basic Graph Coloring -> Ramsey's Theorem for two color (e.g. The party problem for R(3,3) = 6) -> Asymmetric Ramsey Number -> Multicolor Ramsey's Theorem (e.g., Proving R(3,3,3) ≤ 17)).
weight: 5
axis: Instruction Following

criterion: The response explicitly labels each problem as a distinct problem (e.g., Use a section title such as "2.5: Proving R(3,3,3) ≤ 17", "Section 3: Simple Pigeonhole Principle Problem", "The Party Problem", "Chapter 4: Multicolor Ramsey's Theorem").
weight: 3
axis: Communication Quality

criterion: The response ensures that each problem is followed by intuitive reasoning toward the solution (e.g., to show that there are at least two people who have the same number of friends within a group of people >= 2, we have to consider what the holes are and what the pigeons are. The n people are the pigeons that need to be sorted. As for the holes, we need to first consider the fact that the situation where a person has 0 friends and where a friend has n-1 friends is mutually exclusive. Hence, there can only be {0, 1, ...,n-2} or {1, 2,...,n-1} cases, and since there are at most n-1 holes, two people with the same number of friends).
weight: 5
axis: Explicit Criteria

criterion: The response applies the Pigeonhole Principle correctly in at least two different problem contexts (e.g., the Simple N items in N-1 holes problem, the party problem, to show that there are at least two people who have the same number of friends within a group of people >= 2, Five points in an equilateral triangle).
weight: 4
axis: Implicit Criteria

criterion: The response accurately defines Ramsey number R(s,t) in graph-theoretic terms (i.e., Ramsey's theorem states that one will find monochromatic cliques in a sufficiently large complete graph with any edge labeling with colors. The numbers R(s, t) in Ramsey's theorem define the Ramsey number).
weight: 5
axis: Explicit Criteria

criterion: The response correctly proves R(3,3) ≤ 6 using the pigeonhole principle and vertex argument (i.e., Use the party problem where there is a group of people and we wish to find the minimum size of this group so that there is either 3 mutual strangers or 3 mutual friends in the group. Prove the upper bound R(3,3) <= 6 using the pigeonhole principle on a complete graph with 6 vertices, showing that at least 3 edges incident to any of the vertices must be the same color. Now, if we follow those three edges to the tree's other vertices and consider the color of the edges between these 3 vertices, we will find that there exists a monochromatic K3.). 
weight: 3
axis: Implicit Criteria

criterion: The response provides a constructive 2-coloring of K₅ to prove R(3,3) > 5 (i.e., We can construct a counterexample where all the edges wrapping the vertices are of one color and the inner diagonals are the other color. Here, there are no monochromatic triangles.).
weight: 3
axis: Implicit Criteria

criterion: The response explains the proof strategy for R(3,3) in a way accessible to Olympiad-level students (i.e., using simple theorems and if advanced theorems are used, they must be built up intuitively, starting from simpler theorems step-by-step).
weight: 4
axis: Instruction Following

criterion: The response extends to at least one non-symmetric Ramsey number (e.g., R(3,4), R(2,3), R(4,6), R(4,5)).
weight: 3
axis: Implicit Criteria

criterion: The response correctly uses the recurrence inequality R(s,t) ≤ R(s–1,t) + R(s,t–1).
weight: 3
axis: Implicit Criteria

criterion: The response uses at least one problem involving geometric reasoning or spatial partitioning (e.g., Ramsey number on the 2D plane, partitioning a unit cube into regions of different coloring).
weight: 3
axis: Implicit Criteria

criterion: The response includes a multi-color problem with correct reduction to a 2-color case (e.g., Proving R(3,3,3) using R(3,3)).
weight: 2
axis: Implicit Criteria

criterion: The response provides intuition for generalization from 2-color to k-color Ramsey numbers (i.e., the generalization goes from coloring each edge with either of 2 colors (red or blue) to k different colors).
weight: 2
axis: Implicit Criteria

criterion: The response describes a counterexample graph to prove a lower bound (e.g., constructive 2-coloring of K₅ to prove R(3,3) > 5). 
weight: 3
axis: Implicit Criteria

criterion: The response demonstrates a real-world reflection on the meaning of Ramsey theory (e.g., Applications to Computational Geometry, Theoretical Computer Science, Approximation Algorithm for Vertex Cover, and Complexity of Parallel Sorting).
weight: 4
axis: Implicit Criteria

criterion: The response defines basic graph-theoretic concepts clearly (i.e., uses basic explanations of the necessary background material such as graph coloring, cliques, and monochromatic subgraphs).
weight: 4
axis: Implicit Criteria

criterion: The response uses diagrams to explain the problems and solutions (e.g., (1) Diagram of the 2 coloring of K5 to show the counterexample for R(3,3) > 5, (2) Figure to show an example of the Pigeonhole Principle, (3) Drawing graphs for the party problem setup, (4) Drawing graphs for social network problem with friendship edges). 
weight: 4
axis: Communication Quality

criterion: The response clearly separates problem statements from intuitive discussion in structure (i.e., the problem statement is properly formatted and separated from the proof structure or explanation. This includes stating theorems and lemmas in separate boxes, for example).
weight: 2
axis: Communication Quality

criterion: The response reads like an essay instead of a progression of Olympiad problems (i.e., the response should be structured in terms of problem statements, intuitive explanations, proofs, and other mathematical statements such as propositions, theorems, lemmas, and corollaries).
weight: -4
axis: Instruction Following

criterion: The response fails to consistently tie intuition back to how a student might solve each problem (e.g., (1) Messy explanations, (2) Unclear explanations, (3) Irrelevant explanations, (4) No intermediate step to understand how the intuition ties back to the problem).
weight: -4
axis: Communication Quality

criterion: The response contains at least one step making a conceptual leap without adequate problem-level scaffolding (e.g., missing out key steps such as the fact that a vertex cannot have 0 degree and an n-1 degree at the same time, limiting the n options to n-1 holes in the social network problem of proving that at least two friends have the same number of friends in a group).
weight: -3
axis: Communication Quality

criterion: The response uses highly technical machinery without connecting it to problem-solving growth (e.g., (1) Uses the R(3,3) = 6 fact to solve the R(3,3,3) <= 17 without explaining where R(3,3) = 6 came from. (2) Tries to prove the R(3,3) = 6 problem without any explanation of the simple Pigeonhole principle).
weight: -3
axis: Instruction Following

criterion: The response contains messy, unnecessary formulas, code, or tables. (e.g., (1) unrendered latex, (2) messy code snippets with improperly formatted structure or messy comments, (3) broken tables that are unrendered, (4) missing figures or formulas or sections that are referenced in the text).
weight: -4
axis: Communication Quality