A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a t-$(n,k,\lambda )$ combinatorial design with tiny, yet positive, probability.

The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a t-$(n,k,\lambda )$ design exist but, in fact, with positive probability there exists a large set of such designs — that is, a partition of the set of k-subsets of $[n]$ into t-$(n,k,\lambda )$ designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large n, large sets of t-$(n,k,\lambda )$ designs exist whenever $k>12t$ and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all $k>12t$.

Kuperberg，Lovett和Peled引入了一种新的概率技术，用于确定某些规则组合结构的存在（STOC 2012）。使用这种技术，它可以证明在某些条件下，随机选择的结构具有所要求的性质吨-$（ñ，ķ，\lambda ）$ 组合设计的可能性很小，但肯定。

KLP的证明方法适用于显示大量设计和类似组合结构的存在，如下所示。我们修改了随机选择，表明了分析，在同等条件下，不仅一个牛逼-$（ñ，ķ，\lambda ）$设计存在，但实际上，存在大量此类设计的可能性–即，k个子集的一个分区$[ñ]$到Ť -$（ñ，ķ，\lambda ）$设计。具体而言，使用概率方法导出本文中，我们证明了对所有足够大Ñ，大套吨-$（ñ，ķ，\lambda ）$ 设计随时存在 $ķ>12Ť$并满足必要的除数条件。这解决了所有人都可以使用的大型设计方案的存在猜想。$ķ>12Ť$。