Joel Spencer

Probabilistic Methods in Combinatorics

In 1947 Paul Erdős [8] began what is now called the probabilistic method. He showed that if \(\left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array} } \right){{2}^{{1 - \left( {\begin{array}{*{20}{c}} k \\ 2 \\ \end{array} } \right)}}} n.) In modern lanuage he considered the random graph G(n,.5) as described below. For each k-set S let BS denote the “bad” events that S is either a clique or an independent set. Then Pr[BS] = 21-(k/2) so that ΣPr[BS] < 1, hence ∧\( \wedge {\bar B_s}\) ≠ ∅ and a graph satisfying ∧\( \wedge {\bar B_s}\) must exist.

Sudden Emergence of a Giantk-Core in a Random Graph

Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdo?s?R?nyi random graphG(n,?m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork?3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=min?>0?/?k(?) and?k(?)=P{Poisson(?)?k?1}. In particular,c3?3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ?pk(?k)n. Here?kis the minimum point of?/?k(?) andpk(?k)=P{Poisson(?k)?k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one.

Explosive Percolation in Random Networks

Networks in which the formation of connections is governed by a random process often undergo a percolation transition, wherein around a critical point, the addition of a small number of connections causes a sizable fraction of the network to suddenly become linked together. Typically such transitions are continuous, so that the percentage of the network linked together tends to zero right above the transition point. Whether percolation transitions could be discontinuous has been an open question. Here, we show that incorporating a limited amount of choice in the classic Erdös-Rényi network formation model causes its percolation transition to become discontinuous.

Asymptotic lower bounds for Ramsey functions

Abstract A probability theorem, due to Lovasz, is used to derive lower bounds for various Ramsey functions. A short proof of the known result R(3, t) ⩾ ct 2 ( ln t) 2 is given.

Coping with errors in binary search procedures

Abstract We consider the problem of identifying an unknown value x ϵ {1, 2,…, n } using only comparisons of x to constants when as many as E of the comparisons may receive erroneous answers. For a continuous analogue of this problem we show that there is a unique strategy that is optimal in the worst case. This strategy for the continuous problem is then shown to yield a strategy for the original discrete problem that uses log 2 n + E · log 2 log 2 n + O ( E · log 2 E ) comparisons in the worst case. This number is shown to be optimal even if arbitrary “Yes-No” questions are allowed. We show that a modified version of this search problem with errors is equivalent to the problem of finding the minimal root of a set of increasing functions. The modified version is then also shown to be of complexity log 2 n + E · log 2 log 2 n + O ( E · log 2 E ).

Lopsided Lova´sz Local Lemma and Latin transversals

A new version of the Lovasz Local lemma is used to prove the existence of Latin transversals in matrices where no symbol appears too often.

Asymptotic behavior of the chromatic index for hypergraphs

Abstract We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k ), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rodl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to n 2 , resolving a long-standing conjecture.

Sharp concentration of the chromatic number on random graphs G n,p

The distribution of the chromatic number on random graphsGn, p is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity arbitrarily slowly. If the average degreepn is less thann1/6, it concentrates almost surely in five consecutive integers. Large deviation estimates for martingales are used in the proof.

Birth control for giants

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory.We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v1, v2}, {v3, v4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time tc so that S(t) → ∞ as t approaches tc from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < tc. Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > tc we show the existence of a giant component, so that t = tc may be fairly considered a phase transition.Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component.

Ramsey's theorem-A new lower bound

Abstract This paper gives improved asymptotic lower bounds to the Ramsey function R(k, t). Section 1 considers the symmetric case k = t while the more general case is considered in Section 2.