2.2 Generalizations of Ramsey's Theorem

Ramsey's Theorem for hypergraphs extends the classic graph version to higher dimensions. It guarantees the existence of monochromatic substructures in large enough hypergraphs, no matter how the hyperedges are colored.

Generalized Ramsey numbers quantify the minimum size needed to ensure these monochromatic substructures. While their existence is proven, determining exact values or tight bounds remains a challenging open problem in combinatorics.

Ramsey's Theorem Generalizations

Ramsey's theorem for hypergraphs

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• Ramsey's Theorem for graphs establishes for any positive integers $r$ and $s$, there exists a minimum positive integer $R(r,s)$ such that any graph on at least $R(r,s)$ vertices contains either a clique of size $r$ (complete subgraph where every pair of vertices is connected by an edge) or an independent set of size $s$ (set of vertices with no edges between them)
• Ramsey's Theorem for hypergraphs extends this concept, stating for any positive integers $r_1, r_2, ..., r_k$ and any integer $k \geq 2$, there exists a minimum positive integer $R(r_1, r_2, ..., r_k)$ such that any $k$-uniform hypergraph (hypergraph where each hyperedge contains exactly $k$ vertices) on at least $R(r_1, r_2, ..., r_k)$ vertices contains a monochromatic complete $k$-uniform subhypergraph (subhypergraph where all hyperedges of size $k$ are present and have the same color) of size $r_i$ in color $i$ for some $1 \leq i \leq k$
• Key differences between graph and hypergraph versions:
  • Graph version considers edges between pairs of vertices, while hypergraph version considers hyperedges connecting $k$ vertices
  • Graph version guarantees existence of either a clique or an independent set, while hypergraph version guarantees existence of a monochromatic complete $k$-uniform subhypergraph

Arrow notation and Ramsey numbers

• Arrow notation $a_1 \rightarrow (a_2, ..., a_k)^r$ means for any $r$-coloring (assignment of one of $r$ colors to each $k$-tuple of elements) of the $k$-tuples of a set $S$ with $|S| = a_1$, there exists a monochromatic subset (subset where all $k$-tuples have the same color) $A \subseteq S$ with $|A| = a_i$ for some $2 \leq i \leq k$
• Generalized Ramsey number $R(a_2, ..., a_k; r)$ is the smallest positive integer $a_1$ such that $a_1 \rightarrow (a_2, ..., a_k)^r$ holds, representing the minimum size of set $S$ that guarantees existence of a monochromatic subset of size $a_i$ for some $2 \leq i \leq k$ in any $r$-coloring of the $k$-tuples of $S$

Existence of generalized Ramsey numbers

• Theorem states for any positive integers $a_2, ..., a_k$ and $r$, the generalized Ramsey number $R(a_2, ..., a_k; r)$ exists
• Proof by induction on $k$:
  1. Base case ($k = 2$): Existence of $R(a_2; r)$ follows from existence of classical Ramsey numbers for graphs
  2. Inductive step: Assume existence of $R(a_2, ..., a_{k-1}; r)$ for some $k \geq 3$, then prove existence of $R(a_2, ..., a_k; r)$
     • Let $S$ be a set with $|S| = R(R(a_2, ..., a_{k-1}; r), a_k; r)$
     • Consider an arbitrary $r$-coloring of the $k$-tuples of $S$
     • By definition of $R(R(a_2, ..., a_{k-1}; r), a_k; r)$, there exists a monochromatic subset $A \subseteq S$ with either $|A| = R(a_2, ..., a_{k-1}; r)$ or $|A| = a_k$
     • If $|A| = a_k$, proof is complete. If $|A| = R(a_2, ..., a_{k-1}; r)$, then by inductive hypothesis, there exists a monochromatic subset $B \subseteq A$ with $|B| = a_i$ for some $2 \leq i \leq k-1$
• Therefore, $R(a_2, ..., a_k; r)$ exists for all $k \geq 2$

Growth rate of Ramsey numbers

• Classical Ramsey numbers $R(r,s)$ grow exponentially as $r$ and $s$ increase
  • Lower bound: $R(r,s) \geq \binom{r+s-2}{r-1}$ for $r,s \geq 2$
  • Upper bound: $R(r,s) \leq \binom{r+s-2}{r-1} + 1$ for $r,s \geq 2$
• Growth rate of generalized Ramsey numbers $R(a_2, ..., a_k; r)$ is not well-understood, with many open problems related to their bounds
  • Lower bound: $R(a_2, ..., a_k; r) \geq 2^{c_1 \cdot a_2 \cdot \log^{(k-2)} a_2}$ for some constant $c_1 > 0$ and sufficiently large $a_2, ..., a_k$, where $\log^{(k-2)} a_2$ denotes the iterated logarithm (logarithm applied $k-2$ times to $a_2$)
  • Upper bound: $R(a_2, ..., a_k; r) \leq 2^{c_2 \cdot a_2^{k-1}}$ for some constant $c_2 > 0$ and sufficiently large $a_2, ..., a_k$
• Vast gap between lower and upper bounds for generalized Ramsey numbers illustrates difficulty in understanding their precise growth rate