Combinatorica最新文献

A family (mathcal {F} subset mathcal {P}(n)) is r-wise k-intersecting if (|A_1 cap dots cap A_r| ge k) for any (A_1, dots , A_r in mathcal {F}). It is easily seen that if (mathcal {F}) is r-wise k-intersecting for (r ge 2), (k ge 1) then (|mathcal {F}| le 2^{n-1}). The problem of determining the maximum size of a family (mathcal {F}) that is both (r_1)-wise (k_1)-intersecting and (r_2)-wise (k_2)-intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255–1266, 2019). They proved the surprising result that, for ((r_1,k_1) = (3,1)) and ((r_2,k_2) = (2,32)) then this maximum is at most (2^{n-2}), and conjectured the same holds if (k_2) is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for ((r_1,k_1) = (3,1)) and ((r_2,k_2) = (2,3)) for all n.

We answer a question posed by Gallai in 1969 concerning criticality in Sperner’s lemma, listed as Problem 9.14 in the collection of Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). Sperner’s lemma states that if a labelling of the vertices of a triangulation of the d-simplex (Delta ^d) with labels (1, 2, ldots , d+1) has the property that (i) each vertex of (Delta ^d) receives a distinct label, and (ii) any vertex lying in a face of (Delta ^d) has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For (dle 2), it is not difficult to show that for every facet (sigma ), there exists a labelling with the above properties where (sigma ) is the unique rainbow facet. For every (dge 3), however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai’s question as a corollary. The construction is based on the properties of a 4-polytope which had been used earlier to disprove a claim of Motzkin on neighbourly polytopes.

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph (K_{k,k}) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is (Oleft( n^{2 - 1/k}right) ). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that (k ge d ge 2). A remarkable result of Fox et al. (J. Eur. Math. Soc. (JEMS) 19:1785–1810, 2017) with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of (K_{k,k}) as a subgraph must be (Oleft( n^{2 - 1/d}right) ). This theorem is sharp when (k=d=2), because by design any (K_{2,2})-free graph automatically has VC-dimension at most 2, and there are well-known examples of such graphs with (Omega left( n^{3/2}right) ) edges. However, it turns out this phenomenon no longer carries through for any larger d. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of (K_{k,k}) and VC-dimension at most d is (o(n^{2-1/d})), for every (k ge d ge 3).

Given an integer (k>4) and a graph H, we prove that, assuming P(ne ) NP, the List-k -Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if either every component of H is a path on at most three vertices, or removing the isolated vertices of H leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all (kge 1).

We prove that every properly edge-colored n-vertex graph with average degree at least (32(log 5n)^2) contains a rainbow cycle, improving upon the ((log n)^{2+o(1)}) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least (10^5 k^3 n^{1+1/k}) edges contains a rainbow 2k-cycle, which improves the previous bound (2^{ck^2}n^{1+1/k}) obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.

Let (textbf{G}:=(G_1, G_2, G_3)) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in (textbf{G}) is a triple of edges ((e_1, e_2, e_3)) with (e_iin G_i) for each i and ({e_1, e_2, e_3}) forming a triangle in V. The triples (textbf{G}) not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities ((alpha _1, alpha _2, alpha _3)) such that if (vert E(G_i)vert > alpha _i n^2) for each i and n is sufficiently large, then (textbf{G}) must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu.

The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of essential covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the n-cube must be of size at least (Omega (sqrt{n})). We devise a stronger lower bound method, and show that the size of every essential cover is at least (Omega (n^{0.52})). This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems.

We prove that every additive set A with energy (E(A)ge |A|^3/K) has a subset (A'subseteq A) of size (|A'|ge (1-varepsilon )K^{-1/2}|A|) such that (|A'-A'|le O_varepsilon (K^{4}|A'|)). This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc uv in a tournament T is the set of vertices that form a directed triangle with arc uv. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.

We show that (limsup |E(G)|/|V(G)| = 2.5) over all 4-critical planar graphs G, answering a question of Grünbaum from 1988.