Dhruv Mubayi

A hypergraph extension of Turan's theorem

Abstract Fix l ⩾ r ⩾ 2 . Let H l + 1 ( r ) be the r -uniform hypergraph obtained from the complete graph K l + 1 by enlarging each edge with a set of r - 2 new vertices. Thus H l + 1 ( r ) has ( r - 2 ) l + 1 2 + l + 1 vertices and l + 1 2 edges. We prove that the maximum number of edges in an n -vertex r -uniform hypergraph containing no copy of H l + 1 ( r ) is ( l ) r l r n r + o ( n r ) as n → ∞ . This is the first infinite family of irreducible r -uniform hypergraphs for each odd r > 2 whose Turan density is determined. Along the way, we give three proofs of a hypergraph generalization of Turans theorem. We also prove a stability theorem for hypergraphs, analogous to the Simonovits stability theorem for complete graphs.

The Chromatic Spectrum of Mixed Hypergraphs

Abstract. A mixed hypergraph is a triple ℋ=(X, ?, ?), where X is the vertex set, and each of ?, ? is a list of subsets of X. A strict k-coloring of ℋ is a surjection c:X→{1,…,k} such that each member of ? has two vertices assigned a common value and each member of ? has two vertices assigned distinct values. The feasible set of H is {k: H has a strict k-coloring}. Among other results, we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number 1 or is an interval starting with 1. For the set {s,t} with 2≤s≤t−2, the smallest realization has 2t−s vertices. When every member of ?∪? is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers.

On the Turán number of triple systems

For a family of r-graphs F, the Turan number ex(n, F) is the maximum number of edges in an n vertex r-graph that does not contain any member of F. The Turan density π(F)=limn → ∞ ex(n,F)/(n r) When F is an r-graph, π(F) ≠0, and r > 2, determining π (F) is a notoriously hard problem, even for very simple r-graphs F. For example, when r = 3, the value of π(F) is known for very few (< 10) irreducible r-graphs. Building upon a method developed recently by de Caen and Furedi (J. Combin. Theory Ser. B 78 (2000), 274-276), we determine the Turan densities of several 3-graphs that were not previously known. Using this method, we also give a new proof of a result of Frankl and Furedi (Combinatorica 3 (1983), 341-349) that π(H) =2/9, where H has edges 123,124, 345. Let F (3, 2) be the 3-graph 123, 145, 245, 345, let K-4 be the 3-graph 123,124,234, and let C5 be the 3-graph 123, 234, 345, 451, 512. We prove 4/9 ≤ π(J(3,2)) ≤ ½, π({K-4,C5}) ≤ 10/31 = 0.322581, 0.464 < π(C5) ≤ 2 - √2 < 0.586. The middle result is related to a conjecture of Frankl and Furedi (Discrete Math. 50 (1984) 323-328) that π(K-4) = 2/7 The best known bounds are 2/7≤π(K-4)≤1/3.

Stability theorems for cancellative hypergraphs

A cancellative hypergraph has no three edges A, B, C with AΔB ⊂ C. We give a new short proof of an old result of Bollobas, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3-graph. One of the two forbidden subhypergraphs in a cancellative 3-graph is F 5 = {abc, abd, cde}. For n ≥ 33 we show that the maximum number of triples on n vertices containing no copy of F 5 is also achieved by the balanced complete tripartite 3-graph. This strengthens a theorem of Frankl and Furedi, who proved it for n ≥ 3000. For both extremal results, we show that a 3-graph with almost as many edges as the extremal example is approximately tripartite. These stability theorems are analogous to the Simonovits stability theorem for graphs.

Set systems without a simplex or a cluster

AbstractA d-dimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A k-uniform d-cluster is a collection of d+1 sets of size k with empty intersection and union of size at most 2k.We prove the following result which simultaneously addresses an old conjecture of Chvátal [6] and a recent conjecture of the second author [28]. For d≥2 and ζ >0 there is a number T such that the following holds for sufficiently large n. Let G be a k-uniform set system on [n] ={1,…,n} with ζn<k <n/2−T, and suppose either that G contains no d-dimensional simplex or that G contains no d-cluster. Then |G|≤% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4MaeyOe % I0IaeGymaedabaGaem4AaSMaeyOeI0IaeGymaedaaaGaayjkaiaawM % caaaaa!43EB!

Connectivity and separating sets of cages

\left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right)

Signed Domination in Regular Graphs and Set-Systems

with equality only for the family of all k-sets containing a specific element.In the non-uniform setting we obtain the following exact result that generalises a question of Erdős and a result of Milner, who proved the case d=2. Suppose d≥2 and G is a set system on [n] that does not contain a d-dimensional simplex, with n sufficiently large. Then |G|≤2n−1+Σi=0d−1% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4MaeyOe % I0IaeGymaedabaGaemyAaKgaaaGaayjkaiaawMcaaaaa!420A!