Ramsey Numbers of Berge-Hypergraphs and Related Structures

Abstract

For a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection f : E(G) → E(H) such that for every e ∈ E(G), e ⊆ f (e). Let the Ramsey number R r (BG, BG) be the smallest integer n such that for any 2-edge-coloring of a complete r-uniform hypergraph on n vertices, there is a monochromatic Berge-G subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that R 3 (BK s , BK t ) = s + t − 3 for s, t ≥ 4 and max(s, t) ≥ 5 where BK n is a Berge-K n hypergraph. For higher uniformity, we show that R 4 (BK t , BK t ) = t + 1 for t ≥ 6 and R k (BK t , BK t ) = t for k ≥ 5 and t sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.

Partitioning the Power Set of $[n]$ into $C_k$-free parts

“…A more recent research in this direction is by Axenovich and Gyárfás [1], where Ramsey numbers of Berge-G hypergraphs were studied for several graphs G in colorings of P(n). Ramsey numbers of Berge-G hypergraphs in the uniform case have been investigated also in [5,9].…”

“…Proof. Take a partition Q of [9] 3 into 28 classes, each containing three pairwise disjoint triples -a very special case of Baranyai's theorem [2]. However, we need another property of Q: four of these classes X 1 , X 2 , X 3 , X 4 must form a Steiner triple system.…”

“…However, we need another property of Q: four of these classes X 1 , X 2 , X 3 , X 4 must form a Steiner triple system. Then these can be extended by the nine pairs covered by their triples implying that X 1 ∪ X 2 ∪ X 3 ∪ X 4 covers each pair of [9] exactly once. The existence of these X i -s certainly follows from a much stronger result, stating that [9] 3 can be partitioned into seven Steiner triple systems (but probably there are easier ways to get them).…”

“…Then the X i -s provide four claw-free 3-regular partition classes. Partition classes Y i can be defined by putting together 12 pairs of the remaining 24 classes of Q, they form a double cover of [9]. Next we can define 28 partition classes Z i by the complements of the 28 classes of Q, each of them forms a double cover on [9].…”

“…To prepare, set A i = {i + 1, i + 2, i + 3, i + 6}, B i = {i + 4, i + 5, i + 7, i + 8} with arithmetic mod 9. Then 9 double covers of [9] are defined as Then 2 × 9 double covers of [9] are defined as W i = {A i , C i , D i } and R i = {B i , E i , F i }. Note that U i , W i , R i take care of 18 complementary pairs of sizes 4 and 5.…”