Real powers for positive bases, with the zero-base positive-exponent convention

Depends on

• The natural logarithm as the inverse of the exponential function
• The real exponential function and the number $e$ by a power series

Used by

• An α-narrow graph has a clique or stable set of size at least |V(G)|^1/(2α) Corollary
• Compatible extensions from the finite simple core Corollary
• Every bull-free graph has a clique or stable set of size at least |V(G)|^1/4 Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• Lyapunov central limit theorem Corollary
• The edge-density form of Rödl's theorem: every nonempty H-free graph has a linearly large set of self-density at most ε or at least 1-ε Corollary
• The polynomial Rödl property implies the Erdős–Hajnal property Corollary
• A tau-critical graph Definition
• An α-narrow graph Definition
• Euler's real Beta integral Definition
• Hausdorff content at a prescribed scale Definition
• The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class Definition
• The function space Lᵖ(μ) for 0 < p < ∞ Definition
• The polynomial Rödl property for a finite forbidden family Definition
• The real Gamma function by Euler's integral Definition
• The viral property for a finite forbidden family Definition
• A family containing K₁ is viral for vacuous reasons Example
• Every hereditary graph class of bounded order has the Erdős–Hajnal property Example
• Nonidentical strong law under summable normalized variances Example
• Strong law estimator of an integrable mean Example
• Tightness from a uniform moment bound Example
• xˣ tends to one as x tends to zero from the right Example
• A large Y-part in a structural comb partition yields the clique-or-stable-set outcome Lemma
• A wide integral geometric layer forces the complete-or-anticomplete property-(*) blockade Lemma
• Absolute real powers are Borel measurable and convex Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• If G has fewer than (δ n)ʰ induced copies of H and |W|≥λ n, then G[W] has fewer than ((δ/λ)|W|)ʰ Lemma
• If ε is an Erdős–Hajnal constant for H and W is a nonempty vertex set with |W|^ε>hom(G), then G[W] has an induced copy of H Lemma
• The p-functional need not be a norm for 0 < p < 1 Proposition
• A tau-critical graph has no wide pure blockade with cograph pattern Theorem
• Alon–Pach–Solymosi: if H₁ and H₂ have the Erdős–Hajnal property, so does the graph obtained from H₁ by substituting H₂ for a vertex Theorem
• An entire function of polynomial growth is a polynomial Theorem
• An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
• Change of base and inversion of the positive-base real exponential Theorem
• Continuity and derivatives of positive-base real powers Theorem
• Every finite family with the Erdős–Hajnal property is viral Theorem
• Finite variance logarithmic rate for iid sums Theorem
• For 0 < p < infinity, the layer-cake formula computes the integral of |f|ᵖ from the distribution function Theorem
• For every t≥1, the class of Kₜ-free graphs has the Erdős–Hajnal property Theorem

…and 18 more results.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)