Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p e . Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive-decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between Opk log kq and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [GNS17].

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We prove a generalization of Turán's theorem proposed by Balogh and Lidický.

For a positive integer 𝑡, let 𝐹 𝑡 denote the graph of the 𝑡 × 𝑡 grid. Motivated by a 50-year-old conjecture of Erdős about Turán numbers of 𝑟-degenerate graphs, we prove that there exists a constant 𝐶 = 𝐶(𝑡) such that ex(𝑛, 𝐹 𝑡 ) ⩽ 𝐶𝑛 3∕2 . This bound is tight up to the value of 𝐶. One of the interesting ingredients of our proof is a novel way of using the tensor power trick.

Given an r-edge-coloured complete graph Kn, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an r-edge-coloured random graph G(n, p).Recently, Bucić, Korándi and Sudakov established a connection between this problem and a certain Hellytype local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3-colour case by showing that (log n/n) 1/4 is a threshold at which point three monochromatic components are needed to cover all vertices of a 3-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonça, Mota and Schülke. Our approach also allows us to determine the answer in the general r-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi and Sudakov.