Large girth approximate Steiner triple systems

Abstract

In 1973, Erdős asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g 4 for which some g-element vertex-set contains at least g − 2 triples.)We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed 4, we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6 − o(1))n 2 triples and girth larger than . The process iteratively adds random triples subject to the constraint that the girth remains larger than . Our result is best possible up to the o(1)-term, which is a negative power of n.

Degenerate Turán densities of sparse hypergraphs

“…Recent results attained towards resolving this conjecture were proved independently by Bohman and Warnke [8], and Glock, Kühn, Lo and Osthus [27], who showed that for fixed e, there exist e-sparse (n, 3, 2)-packings with size (1 − o(1))n 2 /6, which is near-optimal. A generalization of Conjecture 10 was made by Füredi and Ruszinkó [22] (see also Conjecture 7.2 of [26] for another generalization), who conjectured the existence of e-sparse (n, r, 2)-Steiner systems, where an (n, r, 2)-Steiner system is called e-sparse if it is simultaneously G r (ir − 2i + 2, i)-free for every 2 ≤ i ≤ e.…”

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

“…Answering a question of Erdős [5], two sets of authors in [2] and [8] independently proved that for any ≥ m 2,…”

The Brown–Erdős–Sós conjecture in finite abelian groups

“…Indeed, since (a, b, c, d) is an S 1 -good quadruple, by definition a = c, and therefore b = d, which implies R 1 = R 3 and R 2 = R 4 . Moreover, if any two other quadruples are the same, say R 1 = R 2 , then y (1) 2 = ab 1 = ab = y (2) 2 , contradicting the property that y (1) and y (2) are element-disjoint.…”

“…3 ) ∈ A 3 is element-disjoint from y (1) , and for some a i , b i , c i , d i ∈ A (depending on a, b, c, d), the quadruples 4 ) are all inQ S 0 ( y (1) ), for some S 0 ⊂ S 0 . See Figure 1.…”

“…In order to prove this one should strongly use the group structure, since similar result is not true for general triple systems. In a recent breakthrough result Glock, Kühn, Lo, and Osthus [3], and independently Bohman and Warnke [1], proved a related conjecture of Erdős asymptotically. They proved that there are almost complete Steiner triple systems without containing m points with m − 2 triples.…”