Uniform laws of large numbers form a cornerstone of Vapnik–Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure. Specifically, we assume that the distribution is absolutely continuous with respect to the product of its marginals, a condition that captures many natural settings, including product distributions, sparse mixtures of product distributions, distributions with low mutual information, and more. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite. The linear VC dimension is defined as the maximum size of a shattered set that lies on an axis-parallel line, namely, a set of vectors that agree on all but at most one coordinate. This dimension is always at most the classical VC dimension, yet it can be arbitrarily smaller. For instance, the family of convex sets in ℝ^d has linear VC dimension 2, while its VC dimension is infinite already for d≥ 2. Our proofs rely on estimator that departs substantially from the standard empirical mean estimator and exhibits more intricate structure. We show that such deviations from the standard empirical mean estimator are unavoidable in this setting. Throughout the paper, we propose several open questions, with a particular focus on quantitative sample complexity bounds.

There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already appeared in the literature. Here we systematically study all of these problems solving the new cases exactly or asymptotically. In many cases we also characterize the extremal constructions.

Given a hypergraph with edges of size at most 3, the 3-set cover problem asks to determine the minimum size of a family of edges which covers the vertex set. As the problem is NP-hard, it is natural to consider its fractional (linear programming) relaxation, which provides a lower bound on the value of the optimal solution. The ratio between the actual value and that of the fractional relaxation is called the integrality gap. A classic bound of Lovász implies that the integrality gap in this problem is at most 11/6. This has been improved to 5/3 by Fujito and Okumura. Here we prove that the integrality gap is at most 3/2, which is best possible. A corollary of this result is that the vertex set of any 3-uniform, regular hypergraph on n vertices can be covered by n/2 (or fewer) edges. This solves the k=3 case of a problem of de A. Moreira and Kohayakawa. As another application, we derive a certain variant of the Gale-Shapley stable marriage theorem for triples.

We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely proportionality and maximin share, are not universally feasible, nor are any constant approximations of them. We propose a novel share notion, where an agent assesses the fairness of a bundle by comparing it to her valuation in a random allocation. In this framework, a bundle is considered $q$-quantile fair, for $q\in[0,1]$, if it is at least as good as a bundle obtained in a uniformly random allocation with probability at least $q$. Our main question is whether there exists a constant value of $q$ for which the $q$-quantile share is universally feasible. Our main result establishes a strong connection between the feasibility of quantile shares and the classical Erd\H{o}s Matching Conjecture. Specifically, we show that if a version of this conjecture is true, then the $\frac{1}{2e}$-quantile share is universally feasible. Furthermore, we provide unconditional feasibility results for additive, unit-demand and matroid-rank valuations for constant values of $q$. Finally, we discuss the implications of our results for other share notions.

We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the "larger number game." This generalizes a recent theorem of Hanneke, Livni and Moran. We also propose several applications of our results outside of game theory.

A mechanism guarantees a certain welfare level to its agents, if each of them can secure that level against unanimously adversarial others. How high can such a guarantee be, and what type of mechanism achieves it? In the n‐person probabilistic voting/bargaining model with p deterministic outcomes a guarantee takes the form of a probability distribution over the ranks from 1 to p. If n ≥ p, the uniform lottery is shown to be the only maximal (unimprovable) guarantee. If n < p, combining (variants of) the familiar random dictator and voting by veto mechanisms yields a large family of maximal guarantees: it is exhaustive if n = 2 and almost so if p ≤ 2 n. Voting rules à la Condorcet or Borda, even in probabilistic form, are ruled out by our worst case viewpoint.

We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, Körner and Monti on the maximum number of binary vectors of length n so that every four of them are split into two pairs by some coordinate. We also propose a weaker version of the Erdős–Rado sunflower conjecture.

We prove that a family $\mathcal{T}$ of distinct triangles on $n$ given vertices that does not have a rainbow triangle (that is, three edges, each taken from a different triangle in $\mathcal{T}$, that form together a triangle) must be of size at most $\frac{n^2}{8}$. We also show that this result is sharp and characterize the extremal case. In addition, we discuss a version of this problem in which the triangles are not necessarily distinct, and show that in this case, the same bound holds asymptotically. After posting the original arXiv version of this paper, we learned that the sharp upper bound of $\frac{n^2}{8}$ was proved much earlier by Gy\H{o}ri (2006) and independently by Frankl, F\"uredi and Simonyi (unpublished). Gy\H{o}ri also obtained a stronger version of our result for the case when repetitions are allowed.

We extend the theory of PAC learning in a way which allows to model a rich variety of learning tasks where the data satisfy special properties that ease the learning process. For example, tasks where the distance of the data from the decision boundary is bounded away from zero. The basic and simple idea is to consider partial concepts: these are functions that can be undefined on certain parts of the space. When learning a partial concept, we assume that the source distribution is supported only on points where the partial concept is defined. This way, one can naturally express assumptions on the data such as lying on a lower dimensional surface or margin conditions. In contrast, it is not at all clear that such assumptions can be expressed by the traditional PAC theory. In fact we exhibit easy-to-learn partial concept classes which provably cannot be captured by the traditional PAC theory. This also resolves a question posed by Attias, Kontorovich, and Mansour 2019. We characterize PAC learnability of partial concept classes and reveal an algorithmic landscape which is fundamentally different than the classical one. For example, in the classical PAC model, learning boils down to Empirical Risk Minimization (ERM). In stark contrast, we show that the ERM principle fails in explaining learnability of partial concept classes. In fact, we demonstrate classes that are incredibly easy to learn, but such that any algorithm that learns them must use an hypothesis space with unbounded VC dimension. We also find that the sample compression conjecture fails in this setting. Thus, this theory features problems that cannot be represented nor solved in the traditional way. We view this as evidence that it might provide insights on the nature of learnability in realistic scenarios which the classical theory fails to explain.

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.

A well-known result of Bollobás says that if {(Ai,Bi)}i=1m is a set pair system such that |Ai|≤a and |Bi|≤b for 1≤i≤m, and Ai∩Bj≠0̸ if and only if i≠j, then m≤a+ba. Füredi, Gyárfás and Király recently initiated the study of such systems with the additional property that |Ai∩Bj|=1 for all i≠j. Confirming a conjecture of theirs, we show that this extra condition allows an improvement of the upper bound (at least) by a constant factor.

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.

We prove that any family E1,..., E┌rn┐ of (not necessarily distinct) sets of edges in an r-uniform hypergraph, each having a fractional matching of size n, has a rainbow fractional matching of size n (that is, a set of edges from distinct Ei’s which supports such a fractional matching). When the hypergraph is r-partite and n is an integer, the number of sets needed goes down from rn to rn−r+1. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for r-partite hypergraphs with r>2. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner (arXiv:1805.11278[math.CO]) and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

Lovász proved (see [7]) that given real numbers p1,..., pn, one can round them up or down to integers ϵ1,..., ϵn, in such a way that the total rounding error over every interval (i.e., sum of consecutive pi’s) is at most \(1-\frac{1}{n+1}\). Here we show that the rounding can be done so that for all \(d = 1,...,\left\lfloor {\frac{{n + 1}}{2}} \right\rfloor \), the total rounding error over every union of d intervals is at most \(\left(1-\frac{d}{n+1}\right)d\). This answers a question of Bohman and Holzman [1], who showed that such rounding is possible for each value of d separately.

A famous conjecture of Caccetta and Häggkvist is that in a digraph on n vertices and minimum out-degree at least n r there is a directed cycle of length r or less. We consider the following generalization: in an undirected graph on n vertices, any collection of n disjoint sets of edges, each of size at least n r , has a rainbow cycle of length r or less. We focus on the case r = 3, and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. For any fixed k and large enough n, we determine the maximum number of edges in an n-vertex edge-coloured graph where all colour classes have size at most k and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.

We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with finitely many players and alternatives. No prior knowledge of social choice is assumed: As far as definitions are concerned, the paper is self-contained. Section 2 supplies the necessary general tools for the rest of the paper. Each definition is followed by a simple (but nontrivial) example. In Section 3 we give a complete account of representations of committees (proper and monotonic simple games), by exactly and strongly consistent social choice functions. We start with Peleg’s representations of weak games, and then provide a complete and detailed account of Holzman’s solution of the representation problem for simple games without veto players. In Section 4 we deal with representations of constitutions by game forms. Following Gärdenfors we model a constitution by a monotonic and superadditive effectivity function. We fully characterize the representations for three kinds of equilibrium: Nash equilibrium; acceptable equilibrium (Pareto optimal Nash equilibrium); and strong Nash equilibrium. We conclude in Section 5 with a report on two recent works on representations of constitutions under incomplete information.

Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Holzman and Kleitman (1992) proved that at least 3/8 of these sums satisfy $|S| \le 1$. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13/32, thus breaking the 3/8 barrier.

A d-interval hypergraph has d disjoint copies of the unit interval as its vertex set, and each edge is the union of d subintervals, one on each copy. Extending a classical result of Gallai on the case d = 1, Tardos and Kaiser used topological tools to bound the ratio between the transversal number and the matching number in such hypergraphs. We take a dual point of view, and bound the edge-covering number (namely the minimal number of edges covering the entire vertex set) in terms of a parameter expressing independence of systems of partitions of the d unit intervals. The main tool we use is an extension of the KKM theorem to products of simplices, due to Peleg. Our approach also yields a new proof of the Tardos-Kaiser result.

A network congestion game is played on a directed, two-terminal network. Every player chooses a route from his origin to his destination. The cost of a route is the sum of the costs of the arcs on it. The arc cost is a function of the number of players who use it. Rosenthal proved that such a game always has a Nash equilibrium in pure strategies. Here we pursue a systematic study of the classes of networks for which a strong equilibrium is guaranteed to exist, under two opposite monotonicity assumptions on the arc cost functions. Our main results are: (a) If costs are increasing, strong equilibrium is guaranteed on extension-parallel networks, regardless of whether the players’ origins and destinations are the same or may differ. (b) If costs are decreasing, and the players have the same origin but possibly different destinations, strong equilibrium is guaranteed on series-parallel networks. (c) If costs are decreasing, and both origins and destinations may differ, strong equilibrium is guaranteed on multiextension-parallel networks. In each case, the network condition is not only sufficient but also necessary in order to guarantee strong equilibrium. These results extend and improve earlier ones by Holzman and Law-Yone in the increasing case, and by Epstein et al. in the decreasing case.