69 research outputs found

    The number of subsets of integers with no kk-term arithmetic progression

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    Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of nn, the number of subsets of {1,2,,n}\{1,2,\ldots, n\} that do not contain a kk-term arithmetic progression is at most 2O(rk(n))2^{O(r_k(n))}, where rk(n)r_k(n) is the maximum cardinality of a subset of {1,2,,n}\{1,2,\ldots, n\} without a kk-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of nn, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on rk(n)r_k(n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(rk(n))\Theta(r_k(n)) contain superlinearly many kk-term arithmetic progressions. For integers rr and kk, Erd\Ho s asked whether there is a set of integers SS with no (k+1)(k+1)-term arithmetic progression, but such that any rr-coloring of SS yields a monochromatic kk-term arithmetic progression. Ne\v{s}et\v{r}il and R\"odl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k3k\ge 3 and δ>0\delta>0, there exists a reasonably dense subset of primes SS with no (k+1)(k+1)-term arithmetic progression, yet every USU\subseteq S of size UδS|U|\ge\delta|S| contains a kk-term arithmetic progression. Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.Comment: To appear in International Mathematics Research Notices. This is a longer version than the journal version, containing two additional minor applications of the container metho

    On two problems in Ramsey-Tur\'an theory

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    Alon, Balogh, Keevash and Sudakov proved that the (k1)(k-1)-partite Tur\'an graph maximizes the number of distinct rr-edge-colorings with no monochromatic KkK_k for all fixed kk and r=2,3r=2,3, among all nn-vertex graphs. In this paper, we determine this function asymptotically for r=2r=2 among nn-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an nn-vertex KkK_k-free graph GG with α(G)=o(n)\alpha(G)=o(n). The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page

    The typical structure of maximal triangle-free graphs

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    Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most 2n2/8+o(n2)2^{n^2/8+o(n^2)} nn-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph GG admits a vertex partition XYX\cup Y such that G[X]G[X] is a perfect matching and YY is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint P3P_3's, which is of independent interest.Comment: 17 page

    Proof of Koml\'os's conjecture on Hamiltonian subsets

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    Koml\'os conjectured in 1981 that among all graphs with minimum degree at least dd, the complete graph Kd+1K_{d+1} minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when dd is sufficiently large. In fact we prove a stronger result: for large dd, any graph GG with average degree at least dd contains almost twice as many Hamiltonian subsets as Kd+1K_{d+1}, unless GG is isomorphic to Kd+1K_{d+1} or a certain other graph which we specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ

    Sharp bound on the number of maximal sum-free subsets of integers

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    Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in {1,,n}\{1, \dots , n\} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2n/42^{\lfloor n/4 \rfloor } for the number of maximal sum-free sets. Here, we prove the following: For each 1i41\leq i \leq 4, there is a constant CiC_i such that, given any nimod4n\equiv i \mod 4, {1,,n}\{1, \dots , n\} contains (Ci+o(1))2n/4(C_i+o(1)) 2^{n/4} maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, S\'os and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.Comment: 25 pages, to appear in the Journal of the European Mathematical Societ

    Intersecting families of discrete structures are typically trivial

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    The study of intersecting structures is central to extremal combinatorics. A family of permutations FSn\mathcal{F} \subset S_n is \emph{tt-intersecting} if any two permutations in F\mathcal{F} agree on some tt indices, and is \emph{trivial} if all permutations in F\mathcal{F} agree on the same tt indices. A kk-uniform hypergraph is \emph{tt-intersecting} if any two of its edges have tt vertices in common, and \emph{trivial} if all its edges share the same tt vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for nn sufficiently large with respect to tt, the largest tt-intersecting families in SnS_n are the trivial ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest tt-intersecting kk-uniform hypergraphs are also trivial when nn is large. We determine the \emph{typical} structure of tt-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollob\'as set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira result. Update 2: corrected statement of the unpublished Hamm--Kahn result, and slightly modified notation in Theorem 1.6 Update 3: new title, updated citations, and some minor correction

    Embedding problems and Ramsey-Turán variations in extremal graph theory

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    In this dissertation, we will focus on a few problems in extremal graph theory. The first chapter consists of some basic terms and tools. In Chapter 2, we study a conjecture of Mader on embedding subdivisions of cliques. Improving a bound by Mader, Bollobás and Thomason, and independently Komlós and Szemerédi proved that every graph with average degree d contains a subdivision of K_[Ω(√d)]. The disjoint union of complete bipartite graph K_(r,r) shows that their result is best possible. In particular, this graph does not contain a subdivision of a clique of order w(r). However, one can ask whether their bound can be improved if we forbid such structures. There are various results in this direction, for example Kühn and Osthus proved that their bound can be improved if we forbid a complete bipartite graph of fixed size. Mader proved that that there exists a function g(r) such that every graph G with ẟ(G) ≥ r and girth at least g(r) contains a TK_(r+1). He also asked about the minimum value of g(r). Furthermore, he conjectured that C_4-freeness is enough to guarantee a clique subdivision of order linear in average degree. Some major steps towards these two questions were made by Kühn and Osthus, such as g(r) ≤ 27 and g(r) ≤ 15 for large enough r. In an earlier result, they proved that for C_4-free graphs one can find a subdivision of a clique of order almost linear in minimum degree. Together with József Balogh and Hong Liu, we proved that every C_(2k)-free graph, for k ≥ 3, has such a subdivision of a large clique. We also proved the dense case of Mader's conjecture in a stronger sense. In Chapter 3, we study a graph-tiling problem. Let H be a fixed graph on h vertices and G be a graph on N vertices such that h|n. An H-factor is a collection of n/h vertex-disjoint copies of H in G. The problem of finding sufficient conditions for a graph G to have an H-factor has been extensively studied; most notable is the celebrated Hajnal-Szemerédi Theorem which states that every n-vertex graph G with ẟ(G) ≥ (1-1/r)n has a K_r-factor. The case r=3 was proved earlier by Corrádi and Hajnal. Another type of problems that have been studied over the past few decades are the so-called Ramsey-Turán problems. Erdős and Sós, in 1970, began studying a variation on Turán's theorem: What is the maximum number of edges in an n-vertex, K_r-free graph G if we add extra conditions to avoid the very strict structure of Turán graph. In particular, what if besides being K_r-free, we also require α(G) = o(n) . Since the extremal example for the Hajnal-Szemerédi theorem is very similar to the Turán graph, one can similarly ask how stable is this extremal example. With József Balogh and Theodore Molla, we proved that for an n-vertex graph G with α(G) = o(n), if ẟ(G) ≥ (1/2+o(1))n then G has a triangle factor. This minimum degree condition is asymptotically best possible. We also consider a fractional variant of the Corrádi-Hajnal Theorem, settling the triangle case of a conjecture of Balogh, Kemkes, Lee, and Young. In Chapter 4, we first consider a Ramsey-Turán variant of a theorem of Erd\Ho s. In 1962, he proved that for any r > l ≥ 2, among all K_r-free graphs, the (r-1)-partite Turán graph has the maximum number of copies of K_l. We consider a Ramsey-Turán-type variation of Erdős's result. In particular, we define RT(F,H,f(n)) to be the maximum number of copies of F in an H-free graph with n-vertices and independence number at most f(n). We study this function for different graphs F and H. Recently, Balogh, Hu and Simonovits proved that the Ramsey-Turán function for even cliques experiences a jump. We show that the function RT(K_3,H,f(n)) has a similar behavior when H is an even clique. We also study the sparse analogue of a theorem of Bollobás and Gy\Ho ri about the maximum number of triangles that a C_5-free graph can have. Finally, we consider a Ramsey-Turán variant of a function studied by Erdős and Rothschild about the maximum number of edge-colorings that an n-vertex graph can have without a monochromatic copy of a given graph

    Local conditions for exponentially many subdivisions

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    Given a graph F, let st(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st(F) is exponential in a power of t
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