Abstract We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.

We show that a finitely presented group virtually admits a planar Cayley graph if and only if it is asymptotically minor-excluded, partially answering a conjecture of Georgakopoulos and Papasoglu in the affirmative.

For two graphs F, H and a positive integer n, the function fF,H(n) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where F is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer r and every Kr-1-free graph F, there exists some εF>00$$\\end{document}]]> such that fF,Kr(n)=O(n1/2-εF). This result is tight in two ways. Firstly, it is no longer true if F contains Kr-1 as a subgraph. Secondly, we show that for all r≥4 and ε>00$$\\end{document}]]>, there exists a Kr-1-free graph F for which fF,Kr(n)=Ω(n1/2-ε). Along the way of proving this, we show in particular that for every graph F with minimum degree t, we have fF,K4(n)=Ω(n1/2-6/t). This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants 0<c<C such that for each r≥4, if F is a bipartite graph with sufficiently large minimum degree, then Ω(nclogr)≤fF,Kr(n)≤O(nClogr). This shows that for graphs F with large minimum degree, the behaviour of fF,Kr(n) is drastically different from that of the corresponding off-diagonal Ramsey number fK2,Kr(n).

In the max–min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko (Proceedings of the 38th ACM Symposium on Theory of Computing, 2006). Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell (Graphs Comb 11(3):245–248, 1995) for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods, to replace the combinatorial argument of Haxell (Graphs Comb 11(3):245–248, 1995) for the restricted max–min allocation problem. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. We also study the (1,ε)-restricted version, in which resources can take only two values, and improve the integrality gap in most cases. Our approach applies a criterion of Aharoni and Haxell, and Meshulam, for the existence of independent transversals in graphs, which involves the connectedness of the independence complex. This is complemented by a graph process of Meshulam that decreases the connectedness of the independence complex in a controlled fashion and hence, tailored appropriately to the problem, can verify the criterion. In our applications we aim to establish the flexibility of the approach and hence argue for it to be a potential asset in other optimization problems involving hypergraph matchings.

Given integers n>k>0 k > 0$$\\end{document}]]>, and a set of integers L⊂[0,k-1], an L-system is a family of sets F⊂[n]k such that |F∩F′|∈L for distinct F,F′∈F. L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent to finding the independence number of the graph G(n, k, L). The Lovász numberϑ(G) is a semidefinite programming approximation of the independence number α of a graph G. In this paper, we determine the leading order term of ϑ(G(n,k,L)) of any generalized Johnson graph with k and L fixed and n→∞. As an application of this theorem, we give an explicit construction of a graph G on n vertices with a large gap between the Lovász number and the Shannon capacity c(G). Specifically, we prove that for any ϵ>0 0$$\\end{document}]]>, for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio ϑ(G)/c(G)=Ω(n1-ϵ), which improves on all known constructions. The graph Ga fortiori also has ratio ϑ(G)/α(G)=Ω(n1-ϵ), which greatly improves on the best known explicit construction.