Turan Problem Research Articles

Generalized Turán problems for small graphs

For graphs $H$ and $F$, the generalized Turan number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results in almost all cases, and connect the remaining cases to well-known unsolved problems. Our main new contribution is applying the progressive induction method of Simonovits for generalized Turan problems.

The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm

Let \(\mathcal{T}=(V,\mathcal{E})\) be a 3-uniform linear hypertree. We consider a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\). We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\) of the hypertree \(\mathcal{T}\), with hyperedge densities satisfying some conditions, such that the hypertree \(\mathcal{T}\) does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree \(\mathcal{T}\) in a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\).

The distance to square-free polynomials

In this paper, we consider a variant of Turan's problem on the distance from an integer polynomial in $\mathbb{Z}[x]$ to the nea\-rest irreducible polynomial in $\mathbb{Z}[x]$. We prove that for any polynomial $f \in \mathbb{Z}[x]$, there exist infinitely many square-free polynomials $g\in \mathbb{Z}[x]$ such that $L(f-g) \le 2$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \le 1$. For this, for each integer $d \geq 16$ we construct infinitely many polynomials $f \in \mathbb{Z}[x]$ of degree $d$ such that neither $f$ itself nor any $f(x) \pm x^k$, where $k$ is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.

Some extremal problems for the Fourier transform on the hyperboloid

We give the solution of the Turan, Fejer, Delsarte, Logan, and Bohman extremal problems for the Fourier transform on the hyperboloid ℍ d or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line.

On Two Problems in Ramsey--Turán Theory

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

On the sum necessary to ensure that a degree sequence is potentially H-graphic

Given a graph H, a graphic sequence ź is potentially H-graphic if there is some realization of ź that contains H as a subgraph. In 1991, Erdźs, Jacobson and Lehel posed the following question: Determine the minimum even integer ź(H,n) such that every n-term graphic sequence with sum at least ź(H,n) is potentially H-graphic. This problem can be viewed as a "potential" degree sequence relaxation of the (forcible) Turan problems. While the exact value of ź(H,n) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary H. In this paper, we determine ź(H,n) asymptotically for all H, thereby providing an Erdźs-Stone-Simonovits-type theorem for the Erdźs-Jacobson-Lehel problem.

Some Motzkin–Straus type results for non-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Turan problems of graphs and uniform hypergraphs. Very recently, the study of Turan densities of non-uniform hypergraphs has been motivated by extremal poset problems. Peng et al. showed a generalization of Motzkin---Straus result for $$\{1,2\}$${1,2}-graphs. In this paper, we attempt to explore the relationship between the Lagrangian of a non-uniform hypergraph and the order of its maximum cliques. We give a Motzkin---Straus type result for $$\{1,r\}$${1,r}-graphs. Moreover, we also give an extension of Motzkin---Straus theorem for $$\{1, r_2, \cdots , r_l\}$${1,r2,?,rl}-graphs.

Unavoidable subhypergraphs: a-clusters

Abstract One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a = ( a 1 , … , a p ) be a sequence of positive integers, p ⩾ 2 , k = a 1 + … + a p . An a-cluster is a family of k-sets { F 0 , … , F p } such that the sets F i F 0 are pairwise disjoint ( 1 ⩽ i ⩽ p ) , | F i F 0 | = a i , and the sets F 0 F i are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F 0 F i form an a-partition of F 0 . With an intensive use of the delta-system method we prove that for k > p > 1 and sufficiently large n, ( n > n 0 ( k ) ) , if F is an n-vertex k-uniform family with | F | exceeding the Erdős-Ko-Rado bound ( n − 1 k − 1 ) , then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.

Rainbow Turán Problems

For a fixed graph $H$, we define the rainbow Turan number $\ex^*(n,H)$ to be the maximum number of edges in a graph on $n$ vertices that has a proper edge-colouring with no rainbow $H$. Recall that the (ordinary) Turan number $\ex(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain a copy of $H$. For any non-bipartite $H$ we show that $\ex^*(n,H)=(1+o(1))\ex(n,H)$, and if $H$ is colour-critical we show that $\ex^{*}(n,H)=\ex(n,H)$. When $H$ is the complete bipartite graph $K_{s,t}$ with $s \leq t$ we show $\ex^*(n,K_{s,t}) = O(n^{2-1/s})$, which matches the known bounds for $\ex(n,K_{s,t})$ up to a constant. We also study the rainbow Turan problem for even cycles, and in particular prove the bound $\ex^*(n,C_6) = O(n^{4/3})$, which is of the correct order of magnitude.

Supersaturation For Ramsey-Turán Problems

For an l-graph $${\user1{\mathcal{G}}}$$, the Turan number $${\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}$$ is the maximum number of edges in an n-vertex l-graph $${\user1{\mathcal{H}}}$$ containing no copy of $${\user1{\mathcal{G}}}$$. The limit $$\pi {\left( {\user1{\mathcal{G}}} \right)} = \lim _{{n \to \infty }} {\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}/{\left( {^{n}_{l} } \right)}$$ is known to exist [8]. The Ramsey–Turan density $$p{\left( {\user1{\mathcal{G}}} \right)}$$ is defined similarly to $$\pi {\left( {\user1{\mathcal{G}}} \right)}$$ except that we restrict to only those $${\user1{\mathcal{H}}}$$ with independence number o(n). A result of Erdős and Sos [3] states that $$\pi {\left( {\user1{\mathcal{G}}} \right)} = p{\left( {\user1{\mathcal{G}}} \right)}$$ as long as for every edge E of $${\user1{\mathcal{G}}}$$ there is another edge E′of $${\user1{\mathcal{G}}}$$ for which |E∩E′|≥2. Therefore a natural question is whether there exists $${\user1{\mathcal{G}}}$$ for which $$p{\left( {\user1{\mathcal{G}}} \right)} 2.We prove the existence of a phenomenon similar to supersaturation for Turan problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs $${\user1{\mathcal{G}}}$$ for which $$0 < \ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}$$.We also prove that the 3-graph $${\user1{\mathcal{G}}}$$ with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies $$0 < p{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}$$. The existence of a hypergraph $${\user1{\mathcal{H}}}$$ satisfying $$0 < p{\left( {\user1{\mathcal{H}}} \right)} < \pi {\left( {\user1{\mathcal{H}}} \right)}$$ was conjectured by Erdős and Sos [3], proved by Frankl and Rodl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs.

On the Turan Problem for Periodic Functions with Nonnegative Fourier Coefficients and Small Support

and proved that, for q = 2, 3, . . . , the relation A(1/q) = 1/q holds, the extremal function is φ1/q(x) , and hence derived the asymptotics A(h) = h+O(h) , h → 0 . A. Yu. Popov conjectured that, as h → 0 , the stronger asymptotics A(h) = h+O(h) is valid. Further progress in the Turan problem was made in the papers of Gorbachev and Manoshina [2– 6]. They proved Popov’s conjecture and reduced the Turan problem for rational h to a finitedimensional extremal problem, and evaluated A(h) for h = 2/q, 3/q, 4/q, p/(2p+ 1) . In this paper, the Turan problem is solved for all rational h . We need some results from [6]. Suppose that p, q ∈ N , (p, q) = 1, p ≤ q/2 . Consider the following linear programming problem.

Turán Extremal Problem for Periodic Functions with Small Support and Its Applications

We study a Turan extremal problem on the largest mean value of a 1-periodic even function with nonnegative Fourier coefficients, fixed value at zero, and support on a closed interval \([ - h,h],0 < h \leqslant 1/2\). We show how the solution of this extremal problem for rational numbers h=p/q is related to the solution of two finite-dimensional problems of linear programming. The solution of the Turan problem for rational numbers h of the form 2/q, 3/q, 4/q, \(p/(2p + 1)\) is obtained. Applications of the Turan problem to analytic number theory are given.

Convolution roots of radial positive definite functions with compact support

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| > T admits a representation of the form φ(x) = ∫u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If φ is real-valued and even, can the convolution root u be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of φ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turan's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function φ vanishes outside the unit ball, then ∫|x| 2 f(x) dx = -Δφ(0) ≥ 4j 2 (d-2)/2 where j v denotes the first positive zero of the Bessel function J v , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in R 2 does not exist.

Structural Results About On-line Learning Models With and Without Queries

We solve an open problem of Maass and Turan, showing that the optimal mistake-bound when learning a given concept class without membership queries is within a constant factor of the optimal number of mistakes plus membership queries required by an algorithm that can ask membership queries. Previously known results imply that the constant factor in our bound is best possible. We then show that, in a natural generalization of the mistake-bound model, the usefulness to the learner of arbitrary “yes-no” questions between trials is very limited. We show that several natural structural questions about relatives of the mistake-bound model can be answered through the application of this general result. Most of these results can be interpreted as saying that learning in apparently less powerful (and more realistic) models is not much more difficult than learning in more powerful models.

Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turan concerning the 'distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial P ∈ Z[x] of degree ≤ 22 there exists a monic polynomial Q ∈ Z[x] with deg(Q) = deg(P) such that Q is irreducible over Q and the 'distance' of P and Q is ≤ 4.

What we know and what we do not know about Turán numbers

The numbers which are traditionally named in honor of Paul Turan were introduced by him as a generalization of a problem he solved in 1941. The general problem of Turan having anextremely simple formulation but beingextremely hard to solve, has become one of the most fascinatingextremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.

Applying a result of Frankl and Rödl to the construction of Steiner trees in the hypercube

Let Q( n) be the n-dimensional hypercube, and X a set of points in Q( n). The Steiner problem for the hypercube is to find the smallest number L( n, X) of edges in any subtree of Q( n) which spans X. Let W( k, n) be the set of points in Q( n) having weight k, where we normalize k+1 ⩽ n 2 . We apply a result of Frankl and Rödl on the generalized Turan problem for hypergraphs to show that L(n,W(k+1,n))⩽ n k+1 +(1+ o(1)) log(k) k n k as k→∞ We also show that the second term on the right is within a factor of log( k) from optimal.

Asymptotic solution of the Turán problem for some hypergraphs

Taking an ordinary graphL 2 and replacing each of its vertices by a group ofr/2 vertices (r is even), we get anr-uniform hypergraphL?. In the casex(L 2)=2m+1, we solve asymptotically the Turan problem forL?. Ifx(L 2)#2m+1, the upper bound is still valid but proper constructions are not known.

Lower bounds for Turán's problem

Turan's problem is to determine the maximum numberT(n,k,t) oft-element subsets of ann-set without a complete sub-hypergraph onk vertices, i.e., allt-subsets of ak-set. It is proved that fora?1 fixed andt sufficiently largeT(n, t+a,t)>(1-a(a+4+o(1))logt/( a t )( t n holds

On the lotto problem

The lotto problem generalizes the covering problem and the so-called Turan's problem. We present in this paper original results concerning lower and upper bounds to the “lotto number”.