Large monochromatic components in 3‐edge‐colored Steiner triple systems

Abstract

It is known that in any r‐coloring of the edges of a complete r‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on n vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that (2n+3)/3 is an absolute lower bound and that this lower bound is best possible for infinitely many n. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually (1−o(1))n. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest 3‐partite hole (ie, disjoint sets X1,X2,X3 with false|X1false|=false|X2false|=false|X3false| such that no edge intersects all of X1,X2,X3) in the Steiner triple system (Gyárfás previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the structure of the Steiner triple system and the coloring of its edges are restricted in a certain way. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.

Generalizations and Strengthenings of Ryser's Conjecture

“…Gyárfás also proved that for all r 3 and H ∈ ST S n , mc r (H) n r−1 and this is best possible for infinitely many n when r − 1 ≡ 1, 3 mod 6 and an affine plane of order r − 1 exists. DeBiasio and Tait [21] extended these results showing that, in particular, for almost all H ∈ ST S n , mc 3 (H) (1 − o(1))n. We propose the following problem.…”

Generalizations and strengthenings of Ryser's conjecture

“…Gyárfás also proved that for all r ≥ 3 and H ∈ ST S n , mc r (H) ≥ n r−1 and this is best possible for infinitely many n when r − 1 ≡ 1, 3 mod 6 and an affine plane of order r − 1 exists. DeBiasio and Tait [21] extended these results showing that, in particular, for almost all H ∈ ST S n , mc 3 (H) ≥ (1 − o(1))n. We propose the following problem. Problem 8.11.…”

Large monochromatic components in expansive hypergraphs

“…Let S n be the family of all Steiner triple systems on n vertices. DeBiasio and Tait [6] proved that for all 3-uniform hypergraphs G on n vertices in which every pair of vertices is contained in at least one edge, mc 3 (G) ≥ n − 2α 3 (G) (note that Theorem 1.5(i) is stronger in the sense that there is no requirement that every pair of vertices is contained in at least one edge). They used this to prove that for all S ∈ S n , mc 3 (S) ≥ 2n/3 + 1 and there exists δ > 0 such that for almost all S ∈ S n , mc 3 (S) ≥ n − n 1−δ .…”

“…All of these results for random graphs use the sparse regularity lemma and thus only provide weak bounds on the error terms. Additionally, it was determined in [6] that for almost all Steiner triple systems S on n vertices, mc 3 (S) = (1 − o(1))n. In this case, there is an explicit bound on the error term, but their result is specific to 3 colors and 3-uniform hypergraphs in which every pair of vertices is contained in at least one edge.…”