IA Scholar Query: Transversals in Uniform Hypergraphs with Property (p, 2).
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 01 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Absolutely avoidable order-size pairs in hypergraphs
https://scholar.archive.org/work/b5ujvohxczbh7loikrcmuzz5my
For fixed integer r≥ 2, we call a pair (m,f) of integers, m≥ 1, 0≤ f ≤mr, absolutely avoidable if there is n_0, such that for any pair of integers (n,e) with n>n_0 and 0≤ e≤nr there is an r-uniform hypergraph on n vertices and e edges that contains no induced sub-hypergraph on m vertices and f edges. Some pairs are clearly not absolutely avoidable, for example (m,0) is not absolutely avoidable since any sufficiently sparse hypergraph on at least m vertices contains independent sets on m vertices. Here we show that for any r≥ 3 and m ≥ m_0, either the pair (m, ⌊ mr/2⌋) or the pair (m, ⌊mr/2⌋-m-1) is absolutely avoidable. Next, following the definition of Erdős, Füredi, Rothschild and Sós, we define the density of a pair (m,f) as σ_r(m,f) = lim sup_n →∞|{e : (n,e) → (m,f)}|/ mr. We show that for r≥ 3 most pairs (m,f) satisfy σ_r(m,f)=0, and that for m > r, there exists no pair (m,f) of density 1.Lea Weberwork_b5ujvohxczbh7loikrcmuzz5myMon, 01 Aug 2022 00:00:00 GMTMaximal sets without Choice
https://scholar.archive.org/work/evbb43rq3nenhmpan6qmbn43pe
We show that it is consistent relative to ZF, that there is no well-ordering of ℝ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on ℝ has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either 𝖣𝖢_ω_1 holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projective" can even be replaced with "L(ℝ)". This vastly strengthens earlier consistency results in the literature.Jonathan Schilhanwork_evbb43rq3nenhmpan6qmbn43peMon, 01 Aug 2022 00:00:00 GMTSharp thresholds for Ramsey properties
https://scholar.archive.org/work/ususdnwzwbgatcvw55m6wxi6jm
In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this non-colourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in G_n,p, for all r ≥ 2 and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden's theorem and Schur's theorem, also in any number of colours.Ehud Friedgut, Eden Kuperwasser, Wojciech Samotij, Mathias Schachtwork_ususdnwzwbgatcvw55m6wxi6jmThu, 28 Jul 2022 00:00:00 GMTProblems and results on 1-cross intersecting set pair systems
https://scholar.archive.org/work/lb7m4jyptzbpjhrsoypnlqbttu
The notion of cross intersecting set pair system of size m, ({A_i}_i=1^m, {B_i}_i=1^m) with A_i∩ B_i=∅ and A_i∩ B_j∅, was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that m≤a+b a if |A_i|≤ a and |B_i|≤ b for each i. Our central problem is to see how this bound changes with the additional condition |A_i∩ B_j|=1 for i j. Such a system is called 1-cross intersecting. We show that the maximum size of a 1-cross intersecting set pair system is – at least 5^n/2 for n even, a=b=n, – equal to (⌊n/2⌋+1)(⌈n/2⌉+1) if a=2 and b=n≥ 4, – at most |∪_i=1^m A_i|, – asymptotically n^2 if {A_i} is a linear hypergraph (|A_i∩ A_j|≤ 1 for i j), – asymptotically 1 2n^2 if {A_i} and {B_i} are both linear hypergraphs.Zoltán Füredi, András Gyárfás, Zoltán Királywork_lb7m4jyptzbpjhrsoypnlqbttuSun, 24 Jul 2022 00:00:00 GMTAn Improved Bound for Weak Epsilon-Nets in the Plane
https://scholar.archive.org/work/a4xxlb4nfnbcfh66jpb4q3ac6e
We show that for any finite set P of points in the plane and ϵ>0 there exist O(1/ϵ^3/2+γ) points in ℝ^2, for arbitrary small γ>0, that pierce every convex set K with |K∩ P|≥ϵ |P|. This is the first improvement of the bound of O(1/ϵ^2) that was obtained in 1992 by Alon, Bárány, Füredi and Kleitman for general point sets in the plane.Natan Rubinwork_a4xxlb4nfnbcfh66jpb4q3ac6eThu, 21 Jul 2022 00:00:00 GMTVertex Deletion Parameterized by Elimination Distance and Even Less
https://scholar.archive.org/work/sdzr3cd7lrdmnjd5v32mha6lde
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · n^O(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · n^O(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost k^O(1) in time f(k) · n^O(1). This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczykwork_sdzr3cd7lrdmnjd5v32mha6ldeMon, 18 Jul 2022 00:00:00 GMTQuantum XYZ Product Codes
https://scholar.archive.org/work/yvtu6tdxvbdglktsh3m66sojcu
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3). While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.Anthony Leverrier, Simon Apers, Christophe Vuillotwork_yvtu6tdxvbdglktsh3m66sojcuThu, 14 Jul 2022 00:00:00 GMTXNLP-completeness for Parameterized Problems on Graphs with a Linear Structure
https://scholar.archive.org/work/3tuqcmmmo5gfxlxugcxtc4va6e
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.Hans L. Bodlaender and Carla Groenland and Hugo Jacob and Lars Jaffke and Paloma T. Limawork_3tuqcmmmo5gfxlxugcxtc4va6eWed, 13 Jul 2022 00:00:00 GMTQuantum XYZ Product Codes
https://scholar.archive.org/work/i7z63u6stjdahecwtbsmoooupm
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved. Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N^2/3). While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.Anthony Leverrier, Simon Apers, Christophe Vuillotwork_i7z63u6stjdahecwtbsmoooupmTue, 12 Jul 2022 00:00:00 GMTThresholds for the monochromatic clique transversal game
https://scholar.archive.org/work/hlfhmm3e35ge7m5hqvgwaz2lje
We study a recently introduced two-person combinatorial game, the (a,b)-monochromatic clique transversal game which is played by Alice and Bob on a graph G. As we observe, this game is equivalent to the (b,a)-biased Maker-Breaker game played on the clique-hypergraph of G. Our main results concern the threshold bias a_1(G) that is the smallest integer a such that Alice can win in the (a,1)-monochromatic clique transversal game on G if she is the first to play. Among other results, we determine the possible values of a_1(G) for the disjoint union of graphs, prove a formula for a_1(G) if G is triangle-free, and obtain the exact values of a_1(C_n □ C_m), a_1(C_n □ P_m), and a_1(P_n □ P_m) for all possible pairs (n,m).Csilla Bujtás and Pakanun Dokyeesun and Sandi Klavžarwork_hlfhmm3e35ge7m5hqvgwaz2ljeThu, 07 Jul 2022 00:00:00 GMTExtremal Hypergraph Problems
https://scholar.archive.org/work/v34w6icxbbd67i5we7vcpzjg6u
This thesis studies extremal problems in hypergraph theory, set theory, and graph theory. Results in this thesis can be divided into seven parts. In the first part, we study the feasible region $\Omega(\mathcal{F})$ of a hypergraph family $\mathcal{F}$, which is the set of points $(x,y)$ so that there exists a sequence of $\mathcal{F}$-free $r$-graphs whose shadow densities approach $x$ and whose edge densities approach $y$. We prove some general results about the shape of $\Omega(\mathcal{F})$, and study $\Omega(\mathcal{F})$ for some specific examples such as the cancellative hypergraphs and the expansion of cliques. In the second part, we present a unified framework for proving stability theorems in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph Tur\'{a}n problems to the simpler question of checking that a hypergraph $\mathcal H$ with large minimum degree that omits the forbidden structures is vertex-extendable. We illustrate our method by giving new short proofs to many stability theorems. In the third part, we provide a construction of finite hypergraph families that have arbitrarily (but finite) many extremal configurations. This is the first such construction. Before our work, every family of hypergraphs whose Tur\'{a}n density is known has a unique extremal configuration. In the fourth part, we study problems that are generalizations of the celebrated Erd\H{o}s--Ko--Rado theorem. We give the correct bounds for the size of a family that does not contain a $d$-cluster but contains at least two disjoint edges. This resolves a conjecture of Mammoliti and Britz. We also extend a structural theorem due to Frankl about conditionally intersecting $3$-graphs to the general case, and use it to give new proofs to some theorems in Extremal set theory. Extending the celebrated Katona intersecting shadow theorem, we give tight bounds for the size of the shadow of $t$-intersecting families and families with a bounded matching number. Finally, resolving a conjecture of Muba [...]Xizhi Liuwork_v34w6icxbbd67i5we7vcpzjg6uThu, 07 Jul 2022 00:00:00 GMTCovering random graphs with monochromatic trees
https://scholar.archive.org/work/7ballqmkqzgzhmza4v2o4r7j6i
Given an r-edge-coloured complete graph K_n, 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 Helly-type 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.Domagoj Bradač, Matija Bucićwork_7ballqmkqzgzhmza4v2o4r7j6iTue, 05 Jul 2022 00:00:00 GMTRandom groups do not have Property (T) at densities below 1/4
https://scholar.archive.org/work/qhvn3twp4nfw3njbixqajkx3ju
We prove that random groups in the Gromov density model at density d <1/4 do not have Property (T), answering a conjecture of Przytycki. We also prove similar results in the k-angular model of random groups.Calum J Ashcroftwork_qhvn3twp4nfw3njbixqajkx3juWed, 29 Jun 2022 00:00:00 GMTExact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs
https://scholar.archive.org/work/5iauxvym7re6laatazzwf2bv7u
The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal. In this work, we consider the following model of instances: starting with a set of vertices V, a set S ⊆ V of k vertices is chosen and an arbitrary d-regular bipartite graph is added on it; edges between pairs of vertices in S× (V⧵S) and (V⧵S) × (V⧵S) are added with probability p. Since for d = 0, the problem reduces to recovering a planted independent set, we don't expect efficient algorithms for k = o(√n). This problem is a generalization of the planted balanced biclique problem where the bipartite graph induced on S is a complete bipartite graph; [Yevgeny Levanzov, 2018] gave an algorithm for recovering S in this problem when k = Ω(√n). Our main result is an efficient algorithm that recovers (w.h.p.) the planted bipartite graph when k = Ω_p(√{n log n}) for a large range of parameters. Our results also hold for a natural semi-random model of instances, which involve the presence of a monotone adversary. Our proof shows that a natural SDP relaxation for the problem is integral by constructing an appropriate solution to it's dual formulation. Our main technical contribution is a new approach for construction the dual solution where we calibrate the eigenvectors of the adjacency matrix to be the eigenvectors of the dual matrix. We believe that this approach may have applications to other recovery problems in semi-random models as well. When k = Ω(√n), we give an algorithm for recovering S whose running time is exponential in the number of small eigenvalues in graph induced on S; this algorithm is based on subspace enumeration techniques due to the works of [Alexandra Kolla and Madhur Tulsiani, 2007; Arora et al., 2010; Kolla, 2011].Akash Kumar, Anand Louis, Rameesh Paul, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruffwork_5iauxvym7re6laatazzwf2bv7uTue, 28 Jun 2022 00:00:00 GMTFast winning strategies for Staller in the Maker-Breaker domination game
https://scholar.archive.org/work/u3w4v2lfzndb7avfozfuii53ua
The Maker-Breaker domination game is played on a graph G by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex v ∈ V(G). Dominator's fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant γ'_ SMB(G) (resp., γ_ SMB(G)) which is the smallest integer k such that, under any strategy of Dominator, Staller can win the game by playing at most k vertices, if Staller (resp., Dominator) plays first on the graph G. We prove some basic properties of γ_ SMB(G) and γ'_ SMB(G) and study the parameters' changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality δ(G)+1 ≤γ'_ SMB(G) ≤γ_ SMB(G) always holds and that for every three integers r,s,t with 2≤ r≤ s≤ t, there exists a graph G such that δ(G)+1 = r, γ'_ SMB(G) = s, and γ_ SMB(G) = t. We prove exact formulas for γ'_ SMB(G) where G is a path, or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.Csilla Bujtás, Pakanun Dokyeesunwork_u3w4v2lfzndb7avfozfuii53uaSun, 26 Jun 2022 00:00:00 GMTMatroid-Constrained Maximum Vertex Cover: Approximate Kernels and Streaming Algorithms
https://scholar.archive.org/work/t4h6x4smnnaxbizz3a54cnvqwe
Given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied max k-vertex cover problem, where the matroid is the simple uniform matroid, and it is also a special case of maximizing a monotone submodular function under a matroid constraint. In this work, we give a Fixed Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if k is the rank of the matroid, we obtain (1 - ε) approximation using (1/(ε))^{O(k)}n^{O(1)} time for partition and laminar matroids and using (1/(ε)+k)^{O(k)}n^{O(1)} time for transversal matroids. This extends a result of Manurangsi for uniform matroids [Pasin Manurangsi, 2018]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.Chien-Chung Huang, François Sellier, Artur Czumaj, Qin Xinwork_t4h6x4smnnaxbizz3a54cnvqweWed, 22 Jun 2022 00:00:00 GMTTopos and Stacks of Deep Neural Networks
https://scholar.archive.org/work/xw4jwxtjbbfetawfm7jayjlwgi
Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy of P.Baudot and D.Bennequin in 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators.Jean-Claude Belfiore, Daniel Bennequinwork_xw4jwxtjbbfetawfm7jayjlwgiThu, 16 Jun 2022 00:00:00 GMTTurán density of cliques of order five in 3-uniform hypergraphs with quasirandom links
https://scholar.archive.org/work/qsmwzsp44rg63mtt4bnfxpbo2e
We show that 3-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than 1/3 contain a clique on five vertices. This result is asymptotically best possible.Sören Berger, Simón Piga, Christian Reiher, Vojtěch Rödl, Mathias Schachtwork_qsmwzsp44rg63mtt4bnfxpbo2eWed, 15 Jun 2022 00:00:00 GMTCharacters of local and regular permutation statistics
https://scholar.archive.org/work/7ci6l7ro45d65kdxnnkuzldl6m
We call a function on permutations k-local if it is a linear combination of indicator functions for sets of permutations with fixed values in k positions. Summarizing prior work by Diaconis, Ellis-Friedgut-Pilpel and others, we explain why local functions are best understood as 'low-frequency' with respect to the Fourier transform of the symmetric group. When a k-local function is also a class function, we give an explicit rule for calculating its expansion in the basis {χ^λ} of irreducible 𝔖_n-characters. To do so, we introduce a new path power sum basis {p⃗_μ} of the ring of symmetric functions and prove a Path Murnaghan-Nakayama Rule for the Schur expansion of p⃗_μ. This rule involves novel ribbon tableau combinatorics distinct from the classical Murnaghan-Nakayama Rule and gives a formula for the character sums ∑_w χ^λ(w) over permutations w ∈𝔖_n with fixed values in certain positions. We also introduce a family of local functions we call regular that count weighted occurrences of patterns. Using the theory of character polynomials for the symmetric group, we show regular class functions can be interpreted as multivariate rational functions. This allows us to generalize results of Dimitrov-Khare, Gaetz-Pierson, Gaetz-Ryba on moments for permutation pattern counting statistics and recover a result of Khare-Lorentz-Yan on pattern counting in perfect matchings. We compute conditional expectations for regular statistics applied to uniformly random permutations with a fixed cycle type. As a consequence, we show many regular statistics satisfy a law of large numbers for conjugation invariant probability distributions, depending only on the limiting proportion of fixed points.Zachary Hamaker, Brendon Rhoadeswork_7ci6l7ro45d65kdxnnkuzldl6mWed, 15 Jun 2022 00:00:00 GMTA Product Version of the Hilton-Milner-Frankl Theorem
https://scholar.archive.org/work/aiuzg2y2n5cr7h2castqqqo3kq
Two families ℱ,𝒢 of k-subsets of {1,2,...,n} are called non-trivial cross t-intersecting if |F∩ G|≥ t for all F∈ℱ, G∈𝒢 and |∩{F F∈ℱ}|<t, |∩{G G∈𝒢}|<t. In the present paper, we determine the maximum product of the sizes of two non-trivial cross t-intersecting families of k-subsets of {1,2,...,n} for n≥ 4(t+2)^2k^2, k≥ 5, which is a product version of the Hilton-Milner-Frankl Theorem.Peter Frankl, Jian Wangwork_aiuzg2y2n5cr7h2castqqqo3kqWed, 15 Jun 2022 00:00:00 GMT