Conservation tillage is an extensively used agricultural practice in northern China that alters soil texture and nutrient conditions, causing changes in the soil microbial community. However, how conservation tillage affects rhizosphere and bulk soil fungal communities during plant growth remains unclear. The present study investigated the effect of long-term (6 years) conservation (chisel plow, zero) and conventional (plow) tillage during wheat growth on the rhizosphere fungal community, using high-throughput sequencing of the internal transcribed spacer (ITS) gene and quantitative PCR. During tillering, fungal alpha diversity in both rhizosphere and bulk soil were significantly higher under zero tillage compared to other methods. Although tillage had no significant effect during the flowering stage, fungal alpha diversity at this stage was significantly different between rhizosphere and bulk soils, with bulk soil presenting the highest diversity. This was also reflected in the phylogenetic structure of the communities, as rhizosphere soil communities underwent a greater shift from tillering to flowering compared to bulk soil communities. In general, less variation in community structure was observed under zero tillage compared to plow and chisel plow treatments. Changes in the relative abundance of the fungal orders Capnodiales, Pleosporales, and Xylariales contributed the highest to the dissimilarities observed. Structural equation models revealed that the soil fungal communities under the three tillage regimes were likely influenced by the changes in soil properties associated with plant growth. This study suggested that: (1) differences in nutrient resources between rhizosphere and bulk soils can select for different types of fungi thereby increasing community variation during plant growth; (2) tillage can alter fungal communities' variability, with zero tillage promoting more stable communities. This work suggests that long-term changes in tillage regimes may result in unique soil fungal ecology, which might influence other aspects of soil functioning (e.g., decomposition).

For k ≥ 2 and r ≥ 1 such that k + r ≥ 4, we prove that, for any > 0, there exists > 0 such that the union of an n-vertex k-graph with minimum codegree

For any γ > 0, Keevash, Knox and Mycroft [9] constructed a polynomial-time algorithm which determines the existence of perfect matchings in any n-vertex k-uniform hypergraph whose minimum codegree is at least n/k + γn. We prove a structural theorem that enables us to determine the existence of a perfect matching for any k-uniform hypergraph with minimum codegree at least n/k. This solves a problem of Karpiński, Ruciński and Szymańska completely. Our proof uses a lattice-based absorbing method. 1 2 JIE HAN Construction 1.1 (Space Barrier). Let V be a set of size n and fix S ⊆ V with |S| < n/k. Let H be the k-graph whose edges are all k-sets that intersect S.Note that the minimum codegree of H is |S| and any matching in Construction 1.1 has at most |S| edges.Let PM(k, δ) be the decision problem of determining whether a k-graph H with δ k−1 (H) ≥ δn contains a perfect matching. Given the result of [22], a natural question to ask is the following: For which values of δ can PM(k, δ) be decided in polynomial time? This holds for PM(k, 1/2) by the main result of [22]. On the other hand, PM(k, 0) includes no degree restriction on H at all, so is NP-complete by the result of Karp [6]. Szymańska [23] proved that for δ < 1/k the problem PM(k, δ) admits a polynomial-time reduction to PM(k, 0) and hence PM(k, δ) is also NP-complete. Karpiński, Ruciński and Szymańska showed that there exists ǫ > 0 such that PM(k, 1/2 − ǫ) is in P and asked the complexity of PM(k, δ) for δ ∈ [1/k, 1/2).Recently, Keevash, Knox and Mycroft [9] gave a long and involved proof that shows PM(k, δ) is in P for any δ > 1/k that leaves only PM(k, 1/k) unknown. Moreover, they also constructed a polynomial-time algorithm to find a perfect matching provided one exists. They [8] also expected that it would be difficult to solve the decision problem for δ = 1/k, as n/k is the minimum codegree threshold at which a perfect fractional matching is guaranteed, so there is a clear behavioral change at this point. In this paper, we give a short proof that shows PM(k, δ) is in P for all δ ≥ 1/k and thus solve Problem 1.2 completely. Theorem 1.3. Fix k ≥ 3. Let H be an n-vertex k-graph with δ k−1 (H) ≥ n/k. Then there is an algorithm with running time O(n 3k 2 −5k ), which determines whether H contains a perfect matching.The proof of Theorem 1.3 follows the approach of [9], from which we use several definitions and results. The heart of the algorithm in that paper was a structural theorem [9, Theorem 1.10], which was proved by partitioning the k-graph H into a number of k-partite k-graphs, before finding a perfect matching in each of these k-partite k-graphs by using a theorem of Keevash and Mycroft [10]. Our main improvement is to replace this by a new structural theorem (Theorem 2.6) which significantly simplifies the argument in [9], and which applies in the exact case δ k−1 (H) ≥ n/k (the structural theorem of [9] only applied for δ k−1 (H) ≥ n/k + o(n)). This already provides a polynomial-time algorithm deciding the existence of perfect matchings, and a faster ...

Abstract. The celebrated Erdős-Ko-Rado theorem determines the maximum size of a k-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a k-uniform intersecting family that is not a subfamily of the so-called Erdős-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting k-uniform family that is neither a subfamily of the Erdős-Ko-Rado family nor of the Hilton-Milner family is. For k ≥ 4, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting method, which allows us to solve all cases k ≥ 3 and characterize all extremal families achieving the extremal value.

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